Physics Derivation Graph navigation Sign in
latex:

expression ID Latex list of symbols name notes used in derivation
0000040490 a^2
0000999900 b/(2a)
0001030901 \cos(x)
0001111111 (\sin(x))^2
0001209482 2 \pi
0001304952 \hbar
0001334112 W
0001921933 2 i
0002239424 2
0002338514 \vec{p}_{2}
0002342425 m/m
0002367209 1/2
 
0002393922 x
0002424922 a
0002436656 i \hbar
0002449291 b/(2a)
0002838490 b/(2a)
0002919191 \sin(-x)
0002929944 1/2
0002940021 2 \pi
0003232242 t
0003413423 \cos(-x)
0003747849 -1
0003838111 2
0003919391 x
0003949052 -x
0003949921 \hbar
0003954314 dx
0003981813 -\sin(x)
0004089571 2x
0004264724 y
0004307451 (b/(2a))^2
0004582412 x
0004829194 2
0004831494 a
0004849392 x
0004858592 h
0004934845 x
0004948585 a
0005395034 a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
0005626421 t
0005749291 f
0005938585 \frac{-\hbar^2}{2m}
0006466214 (\sin(x))^2
0006544644 t
0006563727 x
0006644853 c/a
0006656532 e
0007471778 2(\sin(x))^2
0007563791 i
0007636749 x
0007894942 (\sin(x))^2
0008837284 T
0008842811 \cos(2x)
0009458842 \psi(x)
0009484724 \frac{n \pi}{W}x
0009485857 a^2\frac{2}{W}
0009485858 \frac{2n\pi}{W}
0009492929 v du
0009587738 \psi
0009594995 1/2
 
0009877781 y
0203024440 1 = \int_0^W a \sin(\frac{n \pi}{W} x) \psi(x)^* dx
0404050504 \lambda = \frac{v}{f}
0439492440 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\right|_0^W
0934990943 k = \frac{2 \pi}{v T}
0948572140 \int \cos(a x) dx = \frac{1}{a}\sin(a x)
1010393913 \langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*
1010393944 x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
1010923823 k W = n \pi
1020010291 0 = a \sin(k W)
1020394900 p = h/\lambda
1020394902 E = h f
1029039903 p = m v
1029039904 p^2 = m^2 v^2
1158485859 \frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
1202310110 \frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1202312210 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1203938249 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
1248277773 \cos(2x) = 1-2(\sin(x))^2
1293913110 0 = b
1293923844 \lambda = v T
1314464131 \vec{\nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
1314864131 \vec{\nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
1395858355 x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
1492811142 f = x - d  
1492842000 \nabla \vec{x} = f(y)  
1636453295 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = - \nabla^2 \vec{E}
1638282134 \vec{p}_{before} = \vec{p}_{after}
1648958381 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{\nabla}\psi(\vec{r},t)\right)
1857710291 0 = a \sin(n \pi)
1858578388 \nabla^2 E(\vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E(\vec{r})\exp(i \omega t)
1858772113 k = \frac{n \pi}{W}
1934748140 \int |\psi(x)|^2 dx = 1
2029293929 \nabla^2 E(\vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r})\exp(i \omega t)
2103023049 \sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right)
2113211456 f = 1/T
2123139121 -\exp(-i x) = -\cos(x)+i \sin(x)
2131616531 T f = 1
2258485859 {\cal H} \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
2394240499 x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
2394853829 \exp(-i x) = \cos(-x)+i \sin(-x)
2394935831 ( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
2394935835 \left(\langle\psi| \hat{A} |\psi\rangle \right)^+ = \left(\langle a \rangle\right)^+
2395958385 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{-p^2}{\hbar} \psi(\vec{r},t)
2404934990 \langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
2494533900 \langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
2569154141 \vec{\nabla} \times \frac{\partial}{\partial t}\vec{H} = \epsilon_0 \frac{\partial^2 }{\partial t^2}\vec{E}  
2648958382 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p}\psi(\vec{r},t)\right)
2848934890 \langle a \rangle^* = \langle a \rangle
2944838499 \psi(x) = a \sin(\frac{n \pi}{W} x)
3121234211 \frac{k}{2\pi} = \lambda
3121234212 p = \frac{h k}{2\pi}
3121513111 k = \frac{2 \pi}{\lambda}
3131111133 T = 1 / f
3131211131 \omega = 2 \pi f
3132131132 \omega = \frac{2\pi}{T}
3147472131 \frac{\omega}{2 \pi} = f
3285732911 (\cos(x))^2 = 1-(\sin(x))^2
3485475729 \nabla^2 E(\vec{r}) = - \frac{\omega^2}{c^2} E(\vec{r})
3585845894 \langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
3829492824 \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right) = \cos(x)
3924948349 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
3942849294 \exp(i x)-\exp(-i x) = 2 i \sin(x)
3943939590 x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
3947269979 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
3948571256 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi(\vec{r},t)
3948572230 \vec{\nabla}\psi(\vec{r},t) = \psi_0 \vec{\nabla}\exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)  
3948574224 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\vec{k}\cdot\vec{r} - \omega t \right) \right)
3948574226 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
3948574228 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3948574230 \psi(\vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
3948574233 \frac{\partial}{\partial t}\psi(\vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3948574235 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)  
3951205425 \vec{p}_{after} = \vec{p}_{1}
4147472132 E = \frac{h \omega}{2 \pi}
4298359835 E = \frac{1}{2}m v^2
4298359845 E = \frac{1}{2m}m^2 v^2
4298359851 E = \frac{p^2}{2m}
4341171256 i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{p^2}{2 m} \psi(\vec{r},t)
4348571256 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi(\vec{r},t)
4394958389 \vec{\nabla}\cdot \left(\vec{\nabla}\psi(\vec{r},t)\right) = \frac{i}{\hbar} \vec{\nabla}\cdot\left( \vec{p} \psi(\vec{r},t)\right)
4585828572 \epsilon_0 \mu_0 = \frac{1}{c^2}
4585932229 \cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
4598294821 \exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
4638429483 \exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
4742644828 \exp(i x)+\exp(-i x) = 2 \cos(x)
4827492911 \cos(2x)+(\sin(x))^2 = 1-(\sin(x))^2
4838429483 \exp(2 i x) = \cos(2 x)+i \sin(2 x)
4843995999 \frac{1}{2 i}\left(\exp(i x)-\exp(-i x)\right) = \sin(x)
4857472413 1 = \int \psi(x)\psi(x)^* dx
4857475848 \frac{1}{a^2} = \frac{W}{2}
4858429483 \exp(i x)\exp(i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))  
4923339482 i x = \log(y)
4928239482 \log(y) = i x
4928923942 a = b  
4929218492 a+b = c  
4929482992 b = 2
 
4938429482 \exp(-i x) = \cos(x)+i \sin(-x)
4938429483 \exp(i x) = \cos(x)+i \sin(x)
4938429484 \exp(-i x) = \cos(x)-i \sin(x)
4943571230 \vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
4948934890 \langle \psi| \hat{A} |\psi\rangle = \langle a \rangle^*
4949359835 \langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
4954839242 \cos(2x)+i\sin(2x) = (\cos(x)+i \sin(x))(\cos(x)+i \sin(x))  
4978429483 \exp(-i x) = \cos(-x)+i \sin(-x)  
4984892984 a+2 = c
 
4985825552 \nabla^2 E(\vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E(\vec{r})\exp(i \omega t)
5530148480 \vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
5727578862 \frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
5832984291 (\sin(x))^2 + (\cos(x))^2 = 1
5857434758 \int a dx = a x
5868688585 \frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = \frac{p^2}{2m} \psi(\vec{r},t)
5900595848 k = \frac{\omega}{v}
5928285821 x^2 + 2x(b/(2a)) + (b/(2a))^2 = (x+(b/(2a)))^2
5928292841 x^2 + (b/a)x + (b/(2a))^2 = -c/a + (b/(2a))^2
5938459282 x^2 + (b/a)x = -c/a
5958392859 x^2 + (b/a)x+(c/a) = 0
5959282914 x^2 + x(b/a) + (b/(2a))^2 = (x+(b/(2a)))^2
5982958248 x = -\sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
5982958249 x+(b/(2a)) = -\sqrt{(b/(2a))^2 - (c/a)}
5985371230 \vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi(\vec{r},t)
6742123016 \vec{p}_{electron}\cdot\vec{p}_{electron} = (\vec{p}_{1}\cdot\vec{p}_{1})+(\vec{p}_{2}\cdot\vec{p}_{2})-2(\vec{p}_{1}\cdot\vec{p}_{2})
7466829492 \vec{\nabla} \cdot \vec{E} = 0
7564894985 \int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
7572664728 \cos(2x)+2(\sin(x))^2 = 1
7575738420 \left(\sin\left(\frac{n \pi}{W}x\right)\right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
7575859295 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859300 \epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{\nabla} \times \vec{E} )_k = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859302 \epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859304 \epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
7575859306 \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859308 \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859310 \hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859312 \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7917051060 \vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
8139187332 \vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
8257621077 \vec{p}_{before} = \vec{p}_{1}
8311458118 \vec{p}_{after} = \vec{p}_{2}+\vec{p}_{electron}
8399484849 \langle x^2 -2x\langle x \rangle+\langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
8484544728 -a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
8485757728 a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
8485867742 \frac{2}{W} = a^2
8489593958 d(u v) = u dv + v du
8489593960 d(u v) - v du = u dv
8489593962 u dv = d(u v) - v du
8489593964 \int u dv = u v - \int v du
8494839423 \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
8572657110 1 = \int |\psi(x)|^2 dx
8572852424 \vec{E} = E(\vec{r},t)
8575746378 \int \frac{1}{2} dx = \frac{1}{2} x
8575748999 \frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x)\right) = -k^2 \left(a \sin(kx) + b \cos(kx)\right)
8576785890 1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
8577275751 0 = a \sin(0) + b\cos(0)
8582885111 \psi(x) = a \sin(kx) + b \cos(kx)
8582954722 x^2 + 2xh + h^2 = (x+h)^2
8849289982 \psi(x)^* = a \sin(\frac{n \pi}{W} x)
8889444440 1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right)\right)^2 dx
9059289981 \psi(x) = a \sin(k x)
9285928292 ax^2 + bx + c = 0
9291999979 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t}
9294858532 \hat{A}^+ = \hat{A}
9385938295 (x+(b/(2a)))^2 = -(c/a) + (b/(2a))^2
9393939991 \psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9393939992 \psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9394939493 \nabla^2 E(\vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r},t)
9429829482 \frac{d}{dx} y = -\sin(x) + i\cos(x)
9482113948 \frac{dy}{y} = i dx
9482438243 (\cos(x))^2 = \cos(2x)+(\sin(x))^2
9482923849 \exp(i x) = y
9482928242 \cos(2x) = (\cos(x))^2 - (\sin(x))^2
9482928243 \cos(2x)+(\sin(x))^2 = (\cos(x))^2
9482943948 \log(y) = i dx
9482948292 b = c
 
9482984922 \frac{d}{dx} y = (i\sin(x) + \cos(x)) i
9483928192 \cos(2x)+i\sin(2x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
9485384858 \nabla^2 E(\vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E(\vec{r})\exp(i \omega t)
9485747245 \sqrt{\frac{2}{W}} = a
9485747246 -\sqrt{\frac{2}{W}} = a
9492920340 y = \cos(x)+i \sin(x)
9494829190 k  
9495857278 \psi(x=W) = 0
9499428242 E(\vec{r},t) = E(\vec{r})\exp(i \omega t)
9499959299 a + k = b + k  
9582958293 x = \sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
9582958294 x+(b/(2a)) = \sqrt{(b/(2a))^2 - (c/a)}
9584821911 c + d = a  
9585727710 \psi(x=0) = 0
9596004948 x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
9848292229 dy = y i dx
9848294829 \frac{d}{dx} y = y i
9858028950 \frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
9889984281 2(\sin(x))^2 = 1-\cos(2x)
9919999981 \rho = 0
9958485859 \frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
9988949211 (\sin(x))^2 = \frac{1-\cos(2x)}{2}
9991999979 \vec{\nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
9999998870 \frac{\vec{p}}{\hbar} = \vec{k}
9999999870 \frac{p}{\hbar} = k
9999999950 \beta = 1/(k_{Boltzmann}T)  
9999999951 \langle x | k \rangle = \frac{\exp(i k x)}{\sqrt{2\pi}}  
9999999952 f(x) \delta(x-a) = f(a)
 
9999999953 \int_{-\infty}^{\infty} \delta(x) dx = 1
 
9999999954 c = 1/\sqrt{\epsilon_0 \mu_0}  
9999999955 \vec{E} = -\vec{\nabla} \Psi
 
9999999956 \vec{F} = \frac{\partial}{\partial t}\vec{p}  
9999999957 \vec{F} = -\vec{\nabla} V
 
9999999960 \hbar = h/(2 \pi)
9999999961 \frac{E}{\hbar} = \omega
9999999962 p = \hbar k
9999999963 \lambda = h/p
 
9999999964 \omega = c k  
9999999965 E = \omega \hbar
9999999966 \vec{L} = \vec{r}\times\vec{p}
 
9999999967 \vec{S} = \frac{1}{\mu_0} \vec{E}\times \vec{B}
 
9999999968 x = \frac{-b-\sqrt{b^2-4ac}}{2a}
9999999969 x = \frac{-b+\sqrt{b^2-4ac}}{2a}
9999999970 \eta_1 \sin\theta_1 = \eta_2 \sin\theta_2
 
9999999971 {\cal H} = \frac{p^2}{2m}+V
 
9999999972 {\cal H} |\psi\rangle = E |\psi\rangle
 
9999999973 \left( \Delta A \right)^2 = \langle A^2 \rangle - \langle A \rangle^2
 
9999999974 \langle \psi| \hat{A} |\psi\rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx
 
9999999975 \langle \psi| \hat{A} |\psi\rangle = \langle a \rangle
9999999976 \hat{p} = -i \hbar \frac{\partial }{\partial x}  
9999999977 [\hat{x},\hat{p}] = i \hbar  
9999999978 \vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}
 
9999999979 \vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}  
9999999980 \vec{\nabla} \cdot \vec{B} = 0
 
9999999981 \vec{\nabla} \cdot \vec{E} = \rho/\epsilon_0
9999999982 V = I R + Q/C + L \frac{\partial I}{\partial t}  
9999999983 C = V A/d
 
9999999984 Q = C V
 
9999999985 V = I R
 
9999999986 \left[cos⁡θsin⁡θ−sin⁡θcos⁡θ\right]\left[AxAy\right] = \left[Ax′Ay′\right]
 
9999999987 1 atmosphere = 101.325 Pascal
 
9999999988 1 atmosphere = 14.7 pounds/(inch^2)
 
9999999989 mass_{electron} = 511000 electronVolts/(q^2)
 
9999999990 Tesla = 10000*Gauss
 
9999999991 Pascal = Newton / (meter^2)
 
9999999992 Tesla = Newton / (Ampere*meter)
 
9999999993 Farad = Coulumb / Volt
 
9999999995 Volt = Joule / Coulumb
 
9999999996 Coulomb = Ampere / second
 
9999999997 Watt = Joule / second
 
9999999998 Joule = Newton*meter
 
9999999999 Newton = kilogram*meter/(second^2)
 
4961662865 x
9110536742 2 x
8483686863 \sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x)\right)
3470587782 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
8642992037 2
9894826550 2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \left(\exp(i x)+\exp(-i x)\right)
4257214632 \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)  
8699789241 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
9180861128 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
2405307372 \sin(2 x) = 2 \sin(x) \cos(x)
3268645065 x
9350663581 \pi
8332931442 \exp(i \pi) = \cos(\pi)+i \sin(\pi)
6885625907 \exp(i \pi) = -1 + i 0
3331824625 \exp(i \pi) = -1
4901237716 1
2501591100 \exp(i \pi) + 1 = 0
5136652623 E = KE + PE mechanical energy is the sum of the potential plus kinetic energies
7915321076 E, KE, PE  
3192374582 E_2, KE_2, PE_2
 
7875206161 E_2 = KE_2 + PE_2
4229092186 E, KE, PE  
1490821948 E_1, KE_1, PE_1
 
4303372136 E_1 = KE_1 + PE_1
5514556106 E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
8357234146 KE = \frac{1}{2} m v^2 kinetic energy Kinetic_energy
3658066792 KE_2, v_2
 
8082557839 KE, v  
7676652285 KE_2 = \frac{1}{2} m v_2^2
2790304597 KE_1, v_1
 
9880327189 KE, v  
4928007622 KE_1 = \frac{1}{2} m v_1^2
5733146966 KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
6554292307 t
4270680309 \frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
5781981178 x^2 - y^2 = (x+y)(x-y) difference of squares Difference_of_two_squares
5946759357 v_2, v_1
 
6675701064 x, y  
4648451961 v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
9356924046 \frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
9397152918 v = \frac{v_1 + v_2}{2} average velocity
2857430695 a = \frac{v_2 - v_1}{t}
acceleration
7735737409 \frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
4784793837 \frac{KE_2 - KE_1}{t} = m v a
5345738321 F = m a Newton's second law of motion Newton%27s_laws_of_motion#Newton's_second_law
2186083170 \frac{KE_2 - KE_1}{t} = v F
6715248283 PE = -F x potential energy Potential_energy
3852078149 PE_2, x_2
 
7388921188 PE, x  
2431507955 PE_2 = -F x_2
3412641898 PE_1, x_1
 
9937169749 PE, x  
4669290568 PE_1 = -F x_1
7734996511 PE_2 - PE_1 = -F ( x_2 - x_1 )
2016063530 t
7267155233 \frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
9337785146 v = \frac{x_2 - x_1}{t} average velocity
4872970974 \frac{PE_2 - PE_1}{t} = -F v
2081689540 t
2770069250 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
3591237106 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
1772416655 \frac{E_2 - E_1}{t} = v F - F v
1809909100 \frac{E_2 - E_1}{t} = 0
5778176146 t
3806977900 E_2 - E_1 = 0
5960438249 E_1
8558338742 E_2 = E_1
8945218208 \theta_{Brewster} + \theta_{refracted} = 90^{\circ} based on figure 34-27 on page 824 in 2001_HRW
9025853427 \theta_{Brewster}
1310571337 \theta_{refracted} = 90^{\circ} - \theta_{Brewster}
6450985774 n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 ) Law of Refraction eq 34-44 on page 819 in 2001_HRW
1779497048 \theta_{Brewster}, \theta_{refraction}  
7322344455 \theta_1, \theta_2
 
2575937347 n_1 \sin( \theta_{Brewster} ) = n_2 \sin( \theta_{refraction} )
7696214507 n_1 \sin( \theta_{Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{Brewster} )
8588429722 \sin( 90^{\circ} - x ) = \cos( x )
7375348852 \theta_{Brewster}
1512581563 x
6831637424 \sin( 90^{\circ} - \theta_{Brewster} ) = \cos( \theta_{Brewster} )
3061811650 n_1 \sin( \theta_{Brewster} ) = n_2 \cos( \theta_{Brewster} )
7857757625 n_1
9756089533 \sin( \theta_{Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{Brewster} )
5632428182 \cos( \theta_{Brewster} )
2768857871 \frac{\sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )} = \frac{n_2}{n_1}
4968680693 \tan( x ) = \frac{ \sin( x )}{\cos( x )}
7321695558 \theta_{Brewster}
9906920183 x
4501377629 \tan( \theta_{Brewster} ) = \frac{ \sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )}
3417126140 \tan( \theta_{Brewster} ) = \frac{ n_2 }{ n_1 }
5453995431 \arctan{ x }
6023986360 x
8495187962 \theta_{Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
9040079362 f
9565166889 T
5426308937 v = \frac{d}{t}
7476820482 C
1277713901 d
6946088325 v = \frac{C}{t}
6785303857 C = 2 \pi r
4816195267 C, r
 
2113447367 C_{\rm{Earth\ orbit}}, r_{\rm{Earth\ orbit}}
 
6348260313 C_{\rm{Earth\ orbit}} = 2 \pi r_{\rm{Earth\ orbit}}
3847777611 C, v, t  
6406487188 C_{\rm{Earth\ orbit}}, v_{\rm{Earth\ orbit}}, t_{\rm{Earth\ orbit}}
 
3046191961 v_{\rm{Earth\ orbit}} = \frac{C_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
3080027960 v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
7175416299 t_{\rm{Earth\ orbit}} = 1 {\rm{year}}
3219318145 \frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}
8721295221 t_{\rm{Earth\ orbit}} = 3.16 10^7 {\rm{seconds}}
4593428198 v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{3.16\ 10^7 {\rm seconds}}
3472836147 r_{\rm{Earth\ orbit}} = 1.496\ 10^8 {\rm km}
6998364753 v_{\rm{Earth\ orbit}} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
4180845508 v_{\rm{Earth\ orbit}} = 29.8 \frac{{\rm km}}{{\rm sec}}
4662369843 x' = \gamma (x - v t)
2983053062 x = \gamma (x' + v t')
3426941928 x = \gamma ( \gamma (x - v t) + v t' )
2096918413 x = \gamma ( \gamma x - \gamma v t + v t' )
7741202861 x = \gamma^2 x - \gamma^2 v t + \gamma v t'
7337056406 \gamma^2 x
4139999399 x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
3495403335 x
9031609275 x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
8014566709 \gamma^2 v t
9409776983 x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
2226340358 \gamma v
6417705759 \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = \gamma v t'
 
8730201316 \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'
first term was multiplied by \gamma/\gamma
1974334644 \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
5148266645 t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
4287102261 x^2 + y^2 + z^2 = c^2 t^2 describes a spherical wavefront
1201689765 x'^2 + y'^2 + z'^2 = c^2 t'^2
describes a spherical wavefront for an observer in a moving frame of reference
7057864873 y' = y
frame of reference is moving only along x direction
8515803375 z' = z
frame of reference is moving only along x direction
6153463771 3
 
4746576736 asdf
 
1027270623 minga
 
6101803533 ming
 
9231495607 a = b
 
8664264194 b + 2
 
9805063945 \gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2
4057686137 C \to C_{\rm{Earth\ orbit}}
2346150725 r \to r_{\rm{Earth\ orbit}}
9753878784 v \to v_{\rm{Earth\ orbit}}
8135396036 t \to t_{\rm{Earth\ orbit}}
2773628333 \theta_1 \to \theta_{Brewster}
7154592211 \theta_2 \to \theta_{\rm refracted}
3749492596 E \to E_1
1258245373 E \to E_2
4147101187 KE \to KE_1
3809726424 PE \to PE_1
6383056612 KE \to KE_2
5075406409 PE \to PE_2
5398681503 v \to v_1
9305761407 v \to v_2
4218009993 x \to x_1
4188639044 x \to x_2
8173074178 x \to v_2
1025759423 y \to v_1
1935543849 \gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2
1586866563 \left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)
3182633789 \gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
1916173354 -\gamma^2 v^2 + c^2 \gamma^2 = c^2
2076171250 -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0
5284610349 \gamma^2
2417941373 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
5787469164 1 - \gamma^2
1639827492 - c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
5669500954 v^2 \gamma^2
5763749235 -c^2 + c^2 \gamma^2 = v^2 \gamma^2
6408214498 c^2
2999795755 c^2 \gamma^2 = v^2 \gamma^2 + c^2
3412946408 v^2 \gamma^2
2542420160 c^2 \gamma^2 - v^2 \gamma^2 = c^2
7743841045 \gamma^2
7513513483 \gamma^2 (c^2 - v^2) = c^2
8571466509 c^2 - \gamma^2
7906112355 \gamma^2 = \frac{c^2}{c^2 - \gamma^2}
3060139639 1/2
 
1528310784 \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
8360117126 \gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}} not a physically valid result in this context
3366703541 a = \frac{v - v_0}{t} acceleration is the average change in speed over a duration
7083390553 t
4748157455 a t = v - v_0
6417359412 v_0
4798787814 a t + v_0 = v
3462972452 v = v_0 + a t
3411994811 v_{\rm ave} = \frac{d}{t}
6175547907 v_{\rm ave} = \frac{v + v_0}{2}
9897284307 \frac{d}{t} = \frac{v + v_0}{2}
8865085668 t
8706092970 d = \left(\frac{v + v_0}{2}\right)t
7011114072 d = \frac{(v_0 + a t) + v_0}{2} t
1265150401 d = \frac{2 v_0 + a t}{2} t
9658195023 d = v_0 t + \frac{1}{2} a t^2
5799753649 2
7215099603 v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
5144263777 v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
7939765107 v^2 = v_0^2 + 2 a d
6729698807 v_0
9759901995 v - v_0 = a t
2242144313 a
1967582749 t = \frac{v - v_0}{a}
5733721198 d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
5611024898 d = \frac{1}{2 a} (v^2 - v_0^2)
5542390646 2 a
8269198922 2 a d = v^2 - v_0^2
9070454719 v_0^2
4948763856 2 a d + v_0^2 = v^2
9645178657 a t
6457044853 v - a t = v_0
1259826355 d = (v - a t) t + \frac{1}{2} a t^2
4580545876 d = v t - a t^2 + \frac{1}{2} a t^2
6421241247 d = v t - \frac{1}{2} a t^2
4182362050 Z = |Z| \exp( i \theta ) Z \in \mathbb{C}
1928085940 Z^* = |Z| \exp( -i \theta )
9191880568 Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
3350830826 Z Z^* = |Z|^2
8396997949 I = | A + B |^2 intensity of two waves traveling opposite directions on same path
4437214608 Z
5623794884 A + B
2236639474 (A + B)(A + B)^* = |A + B|^2
1020854560 I = (A + B)(A + B)^*
6306552185 I = (A + B)(A^* + B^*)
8065128065 I = A A^* + B B^* + A B^* + B A^*
9761485403 Z
8710504862 A
4075539836 A A^* = |A|^2
6529120965 B
1511199318 Z
7107090465 B B^* = |B|^2
5125940051 I = |A|^2 + B B^* + A B^* + B A^*
1525861537 I = |A|^2 + |B|^2 + A B^* + B A^*
1742775076 Z \to B
7607271250 \theta \to \phi
4192519596 B = |B| \exp(i \phi)
2064205392 A
1894894315 Z
1357848476 A = |A| \exp(i \theta)
6555185548 A^* = |A| \exp(-i \theta)
4504256452 B^* = |B| \exp(-i \phi)
7621705408 I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
3085575328 I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
4935235303 x
2293352649 \theta - \phi
3660957533 \cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
3967985562 2
2700934933 2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
8497631728 I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
6774684564 \theta = \phi for coherent waves
2099546947 I = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )  
2719691582 |A| = |B| in a loop
3257276368 I = |A|^2 + |A|^2 + |A| |A| 2
 
8283354808 I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
8046208134 I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
1172039918 I_{\rm coherent} = 4 |A|^2
8602221482 \langle \cos(\theta - \phi) \rangle = 0 incoherent light source
6240206408 I_{\rm incoherent} = |A|^2 + |B|^2
6529793063 I_{\rm incoherent} = |A|^2 + |A|^2
3060393541 I_{\rm incoherent} = 2|A|^2
6556875579 \frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
8750500372 f
 
7543102525 g
 
8267133829 a = b
 
7968822469 c = d
 
3169580383 \vec{a} = \frac{d\vec{v}}{dt} acceleration is the change in speed over a duration
8602512487 \vec{a} = a_x \hat{x} + a_y \hat{y} decompose acceleration into two components
4158986868 a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
5349866551 \vec{v} = v_x \hat{x} + v_y \hat{y}
7729413831 a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
1819663717 a_x = \frac{d}{dt} v_x
8228733125 a_y = \frac{d}{dt} v_y
9707028061 a_x = 0
2741489181 a_y = -g
3270039798 2
8880467139 2
8750379055 0 = \frac{d}{dt} v_x
1977955751 -g = \frac{d}{dt} v_y
6672141531 dt
1702349646 -g dt = d v_y
8584698994 -g \int dt = \int d v_y
9973952056 -g t = v_y - v_{0, y}
4167526462 v_{0, y}
6572039835 -g t + v_{0, y} = v_y
7252338326 v_y = \frac{dy}{dt}
6204539227 -g t + v_{0, y} = \frac{dy}{dt}
1614343171 dt
8145337879 -g t dt + v_{0, y} dt = dy
8808860551 -g \int t dt + v_{0, y} \int dt = \int dy
2858549874 - \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
6098638221 y_0
2461349007 - \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
8717193282 dt
1166310428 0 dt = d v_x
2366691988 \int 0 dt = \int d v_x
1676472948 0 = v_x - v_{0, x}
1439089569 v_{0, x}
6134836751 v_{0, x} = v_x
8460820419 v_x = \frac{dx}{dt}
7455581657 v_{0, x} = \frac{dx}{dt}
8607458157 dt
1963253044 v_{0, x} dt = dx
3676159007 v_{0, x} \int dt = \int dx
9882526611 v_{0, x} t = x - x_0
3182907803 x_0
8486706976 v_{0, x} t + x_0 = x
1306360899 x = v_{0, x} t + x_0
7049769409 2
9341391925 \vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
6410818363 \theta
7391837535 \cos(\theta) = \frac{v_{0, x}}{v_0}
7376526845 \sin(\theta) = \frac{v_{0, y}}{v_0}
5868731041 v_0
6083821265 v_0 \cos(\theta) = v_{0, x}
5438722682 x = v_0 t \cos(\theta) + x_0
5620558729 v_0
8949329361 v_0 \sin(\theta) = v_{0, y}
1405465835 y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
9862900242 y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
6050070428 v_{0, x}
3274926090 t = \frac{x - x_0}{v_{0, x}}
7354529102 y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0
8406170337 y \to y_f
2403773761 t \to t_f
5379546684 y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
5373931751 t = t_f
9112191201 y_f = 0
8198310977 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
1650441634 y_0 = 0
define coordinate system such that initial height is at origin
1087417579 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
4829590294 t_f
2086924031 0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
6974054946 \frac{1}{2} g t_f
1191796961 \frac{1}{2} g t_f = v_0 \sin(\theta)
2510804451 2/g
4778077984 t_f = \frac{2 v_0 \sin(\theta)}{g}
3273630811 x \to x_f
6732786762 t \to t_f
3485125659 x_f = v_0 t_f \cos(\theta) + x_0
4370074654 t = t_f
2378095808 x_f = x_0 + d
4268085801 x_0 + d = v_0 t_f \cos(\theta) + x_0
8072682558 x_0
7233558441 d = v_0 t_f \cos(\theta)
2297105551 d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
7587034465 \theta
7214442790 x
2519058903 \sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
8922441655 d = \frac{v_0^2}{g} \sin(2 \theta)
5667870149 \theta
1541916015 \theta = \frac{\pi}{4}
3607070319 d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
5353282496 d = \frac{v_0^2}{g}
0000040490 a^2
0000999900 b/(2a)
0001030901 \cos(x)
0001111111 (\sin(x))^2
0001209482 2 \pi
0001304952 \hbar
0001334112 W
0001921933 2 i
0002239424 2
0002338514 \vec{p}_{2}
0002342425 m/m
0002367209 1/2
 
0002393922 x
0002424922 a
0002436656 i \hbar
0002449291 b/(2a)
0002838490 b/(2a)
0002919191 \sin(-x)
0002929944 1/2
0002940021 2 \pi
0003232242 t
0003413423 \cos(-x)
0003747849 -1
0003838111 2
0003919391 x
0003949052 -x
0003949921 \hbar
0003954314 dx
0003981813 -\sin(x)
0004089571 2x
0004264724 y
0004307451 (b/(2a))^2
0004582412 x
0004829194 2
0004831494 a
0004849392 x
0004858592 h
0004934845 x
0004948585 a
0005395034 a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
0005626421 t
0005749291 f
0005938585 \frac{-\hbar^2}{2m}
0006466214 (\sin(x))^2
0006544644 t
0006563727 x
0006644853 c/a
0006656532 e
0007471778 2(\sin(x))^2
0007563791 i
0007636749 x
0007894942 (\sin(x))^2
0008837284 T
0008842811 \cos(2x)
0009458842 \psi(x)
0009484724 \frac{n \pi}{W}x
0009485857 a^2\frac{2}{W}
0009485858 \frac{2n\pi}{W}
0009492929 v du
0009587738 \psi
0009594995 1/2
 
0009877781 y
0203024440 1 = \int_0^W a \sin(\frac{n \pi}{W} x) \psi(x)^* dx
0404050504 \lambda = \frac{v}{f}
0439492440 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\right|_0^W
0934990943 k = \frac{2 \pi}{v T}
0948572140 \int \cos(a x) dx = \frac{1}{a}\sin(a x)
1010393913 \langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*
1010393944 x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
1010923823 k W = n \pi
1020010291 0 = a \sin(k W)
1020394900 p = h/\lambda
1020394902 E = h f
1029039903 p = m v
1029039904 p^2 = m^2 v^2
1158485859 \frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
1202310110 \frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1202312210 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1203938249 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
1248277773 \cos(2x) = 1-2(\sin(x))^2
1293913110 0 = b
1293923844 \lambda = v T
1314464131 \vec{\nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
1314864131 \vec{\nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
1395858355 x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
1492811142 f = x - d  
1492842000 \nabla \vec{x} = f(y)  
1636453295 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = - \nabla^2 \vec{E}
1638282134 \vec{p}_{before} = \vec{p}_{after}
1648958381 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{\nabla}\psi(\vec{r},t)\right)
1857710291 0 = a \sin(n \pi)
1858578388 \nabla^2 E(\vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E(\vec{r})\exp(i \omega t)
1858772113 k = \frac{n \pi}{W}
1934748140 \int |\psi(x)|^2 dx = 1
2029293929 \nabla^2 E(\vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r})\exp(i \omega t)
2103023049 \sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right)
2113211456 f = 1/T
2123139121 -\exp(-i x) = -\cos(x)+i \sin(x)
2131616531 T f = 1
2258485859 {\cal H} \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
2394240499 x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
2394853829 \exp(-i x) = \cos(-x)+i \sin(-x)
2394935831 ( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
2394935835 \left(\langle\psi| \hat{A} |\psi\rangle \right)^+ = \left(\langle a \rangle\right)^+
2395958385 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{-p^2}{\hbar} \psi(\vec{r},t)
2404934990 \langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
2494533900 \langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
2569154141 \vec{\nabla} \times \frac{\partial}{\partial t}\vec{H} = \epsilon_0 \frac{\partial^2 }{\partial t^2}\vec{E}  
2648958382 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p}\psi(\vec{r},t)\right)
2848934890 \langle a \rangle^* = \langle a \rangle
2944838499 \psi(x) = a \sin(\frac{n \pi}{W} x)
3121234211 \frac{k}{2\pi} = \lambda
3121234212 p = \frac{h k}{2\pi}
3121513111 k = \frac{2 \pi}{\lambda}
3131111133 T = 1 / f
3131211131 \omega = 2 \pi f
3132131132 \omega = \frac{2\pi}{T}
3147472131 \frac{\omega}{2 \pi} = f
3285732911 (\cos(x))^2 = 1-(\sin(x))^2
3485475729 \nabla^2 E(\vec{r}) = - \frac{\omega^2}{c^2} E(\vec{r})
3585845894 \langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
3829492824 \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right) = \cos(x)
3924948349 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
3942849294 \exp(i x)-\exp(-i x) = 2 i \sin(x)
3943939590 x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
3947269979 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
3948571256 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi(\vec{r},t)
3948572230 \vec{\nabla}\psi(\vec{r},t) = \psi_0 \vec{\nabla}\exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)  
3948574224 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\vec{k}\cdot\vec{r} - \omega t \right) \right)
3948574226 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
3948574228 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3948574230 \psi(\vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
3948574233 \frac{\partial}{\partial t}\psi(\vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3948574235 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)  
3951205425 \vec{p}_{after} = \vec{p}_{1}
4147472132 E = \frac{h \omega}{2 \pi}
4298359835 E = \frac{1}{2}m v^2
4298359845 E = \frac{1}{2m}m^2 v^2
4298359851 E = \frac{p^2}{2m}
4341171256 i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{p^2}{2 m} \psi(\vec{r},t)
4348571256 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi(\vec{r},t)
4394958389 \vec{\nabla}\cdot \left(\vec{\nabla}\psi(\vec{r},t)\right) = \frac{i}{\hbar} \vec{\nabla}\cdot\left( \vec{p} \psi(\vec{r},t)\right)
4585828572 \epsilon_0 \mu_0 = \frac{1}{c^2}
4585932229 \cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
4598294821 \exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
4638429483 \exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
4742644828 \exp(i x)+\exp(-i x) = 2 \cos(x)
4827492911 \cos(2x)+(\sin(x))^2 = 1-(\sin(x))^2
4838429483 \exp(2 i x) = \cos(2 x)+i \sin(2 x)
4843995999 \frac{1}{2 i}\left(\exp(i x)-\exp(-i x)\right) = \sin(x)
4857472413 1 = \int \psi(x)\psi(x)^* dx
4857475848 \frac{1}{a^2} = \frac{W}{2}
4858429483 \exp(i x)\exp(i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))  
4923339482 i x = \log(y)
4928239482 \log(y) = i x
4928923942 a = b  
4929218492 a+b = c  
4929482992 b = 2
 
4938429482 \exp(-i x) = \cos(x)+i \sin(-x)
4938429483 \exp(i x) = \cos(x)+i \sin(x)
4938429484 \exp(-i x) = \cos(x)-i \sin(x)
4943571230 \vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
4948934890 \langle \psi| \hat{A} |\psi\rangle = \langle a \rangle^*
4949359835 \langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
4954839242 \cos(2x)+i\sin(2x) = (\cos(x)+i \sin(x))(\cos(x)+i \sin(x))  
4978429483 \exp(-i x) = \cos(-x)+i \sin(-x)  
4984892984 a+2 = c
 
4985825552 \nabla^2 E(\vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E(\vec{r})\exp(i \omega t)
5530148480 \vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
5727578862 \frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
5832984291 (\sin(x))^2 + (\cos(x))^2 = 1
5857434758 \int a dx = a x
5868688585 \frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = \frac{p^2}{2m} \psi(\vec{r},t)
5900595848 k = \frac{\omega}{v}
5928285821 x^2 + 2x(b/(2a)) + (b/(2a))^2 = (x+(b/(2a)))^2
5928292841 x^2 + (b/a)x + (b/(2a))^2 = -c/a + (b/(2a))^2
5938459282 x^2 + (b/a)x = -c/a
5958392859 x^2 + (b/a)x+(c/a) = 0
5959282914 x^2 + x(b/a) + (b/(2a))^2 = (x+(b/(2a)))^2
5982958248 x = -\sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
5982958249 x+(b/(2a)) = -\sqrt{(b/(2a))^2 - (c/a)}
5985371230 \vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi(\vec{r},t)
6742123016 \vec{p}_{electron}\cdot\vec{p}_{electron} = (\vec{p}_{1}\cdot\vec{p}_{1})+(\vec{p}_{2}\cdot\vec{p}_{2})-2(\vec{p}_{1}\cdot\vec{p}_{2})
7466829492 \vec{\nabla} \cdot \vec{E} = 0
7564894985 \int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
7572664728 \cos(2x)+2(\sin(x))^2 = 1
7575738420 \left(\sin\left(\frac{n \pi}{W}x\right)\right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
7575859295 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859300 \epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{\nabla} \times \vec{E} )_k = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859302 \epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859304 \epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
7575859306 \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859308 \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859310 \hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859312 \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7917051060 \vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
8139187332 \vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
8257621077 \vec{p}_{before} = \vec{p}_{1}
8311458118 \vec{p}_{after} = \vec{p}_{2}+\vec{p}_{electron}
8399484849 \langle x^2 -2x\langle x \rangle+\langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
8484544728 -a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
8485757728 a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
8485867742 \frac{2}{W} = a^2
8489593958 d(u v) = u dv + v du
8489593960 d(u v) - v du = u dv
8489593962 u dv = d(u v) - v du
8489593964 \int u dv = u v - \int v du
8494839423 \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
8572657110 1 = \int |\psi(x)|^2 dx
8572852424 \vec{E} = E(\vec{r},t)
8575746378 \int \frac{1}{2} dx = \frac{1}{2} x
8575748999 \frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x)\right) = -k^2 \left(a \sin(kx) + b \cos(kx)\right)
8576785890 1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
8577275751 0 = a \sin(0) + b\cos(0)
8582885111 \psi(x) = a \sin(kx) + b \cos(kx)
8582954722 x^2 + 2xh + h^2 = (x+h)^2
8849289982 \psi(x)^* = a \sin(\frac{n \pi}{W} x)
8889444440 1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right)\right)^2 dx
9059289981 \psi(x) = a \sin(k x)
9285928292 ax^2 + bx + c = 0
9291999979 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t}
9294858532 \hat{A}^+ = \hat{A}
9385938295 (x+(b/(2a)))^2 = -(c/a) + (b/(2a))^2
9393939991 \psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9393939992 \psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9394939493 \nabla^2 E(\vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r},t)
9429829482 \frac{d}{dx} y = -\sin(x) + i\cos(x)
9482113948 \frac{dy}{y} = i dx
9482438243 (\cos(x))^2 = \cos(2x)+(\sin(x))^2
9482923849 \exp(i x) = y
9482928242 \cos(2x) = (\cos(x))^2 - (\sin(x))^2
9482928243 \cos(2x)+(\sin(x))^2 = (\cos(x))^2
9482943948 \log(y) = i dx
9482948292 b = c
 
9482984922 \frac{d}{dx} y = (i\sin(x) + \cos(x)) i
9483928192 \cos(2x)+i\sin(2x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
9485384858 \nabla^2 E(\vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E(\vec{r})\exp(i \omega t)
9485747245 \sqrt{\frac{2}{W}} = a
9485747246 -\sqrt{\frac{2}{W}} = a
9492920340 y = \cos(x)+i \sin(x)
9494829190 k  
9495857278 \psi(x=W) = 0
9499428242 E(\vec{r},t) = E(\vec{r})\exp(i \omega t)
9499959299 a + k = b + k  
9582958293 x = \sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
9582958294 x+(b/(2a)) = \sqrt{(b/(2a))^2 - (c/a)}
9584821911 c + d = a  
9585727710 \psi(x=0) = 0
9596004948 x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
9848292229 dy = y i dx
9848294829 \frac{d}{dx} y = y i
9858028950 \frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
9889984281 2(\sin(x))^2 = 1-\cos(2x)
9919999981 \rho = 0
9958485859 \frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
9988949211 (\sin(x))^2 = \frac{1-\cos(2x)}{2}
9991999979 \vec{\nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
9999998870 \frac{\vec{p}}{\hbar} = \vec{k}
9999999870 \frac{p}{\hbar} = k
9999999950 \beta = 1/(k_{Boltzmann}T)  
9999999951 \langle x | k \rangle = \frac{\exp(i k x)}{\sqrt{2\pi}}  
9999999952 f(x) \delta(x-a) = f(a)
 
9999999953 \int_{-\infty}^{\infty} \delta(x) dx = 1
 
9999999954 c = 1/\sqrt{\epsilon_0 \mu_0}  
9999999955 \vec{E} = -\vec{\nabla} \Psi
 
9999999956 \vec{F} = \frac{\partial}{\partial t}\vec{p}  
9999999957 \vec{F} = -\vec{\nabla} V
 
9999999960 \hbar = h/(2 \pi)
9999999961 \frac{E}{\hbar} = \omega
9999999962 p = \hbar k
9999999963 \lambda = h/p
 
9999999964 \omega = c k  
9999999965 E = \omega \hbar
9999999966 \vec{L} = \vec{r}\times\vec{p}
 
9999999967 \vec{S} = \frac{1}{\mu_0} \vec{E}\times \vec{B}
 
9999999968 x = \frac{-b-\sqrt{b^2-4ac}}{2a}
9999999969 x = \frac{-b+\sqrt{b^2-4ac}}{2a}
9999999970 \eta_1 \sin\theta_1 = \eta_2 \sin\theta_2
 
9999999971 {\cal H} = \frac{p^2}{2m}+V
 
9999999972 {\cal H} |\psi\rangle = E |\psi\rangle
 
9999999973 \left( \Delta A \right)^2 = \langle A^2 \rangle - \langle A \rangle^2
 
9999999974 \langle \psi| \hat{A} |\psi\rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx
 
9999999975 \langle \psi| \hat{A} |\psi\rangle = \langle a \rangle
9999999976 \hat{p} = -i \hbar \frac{\partial }{\partial x}  
9999999977 [\hat{x},\hat{p}] = i \hbar  
9999999978 \vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}
 
9999999979 \vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}  
9999999980 \vec{\nabla} \cdot \vec{B} = 0
 
9999999981 \vec{\nabla} \cdot \vec{E} = \rho/\epsilon_0
9999999982 V = I R + Q/C + L \frac{\partial I}{\partial t}  
9999999983 C = V A/d
 
9999999984 Q = C V
 
9999999985 V = I R
 
9999999986 \left[cos⁡θsin⁡θ−sin⁡θcos⁡θ\right]\left[AxAy\right] = \left[Ax′Ay′\right]
 
9999999987 1 atmosphere = 101.325 Pascal
 
9999999988 1 atmosphere = 14.7 pounds/(inch^2)
 
9999999989 mass_{electron} = 511000 electronVolts/(q^2)
 
9999999990 Tesla = 10000*Gauss
 
9999999991 Pascal = Newton / (meter^2)
 
9999999992 Tesla = Newton / (Ampere*meter)
 
9999999993 Farad = Coulumb / Volt
 
9999999995 Volt = Joule / Coulumb
 
9999999996 Coulomb = Ampere / second
 
9999999997 Watt = Joule / second
 
9999999998 Joule = Newton*meter
 
9999999999 Newton = kilogram*meter/(second^2)
 
4961662865 x
9110536742 2 x
8483686863 \sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x)\right)
3470587782 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
8642992037 2
9894826550 2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \left(\exp(i x)+\exp(-i x)\right)
4257214632 \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)  
8699789241 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
9180861128 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
2405307372 \sin(2 x) = 2 \sin(x) \cos(x)
3268645065 x
9350663581 \pi
8332931442 \exp(i \pi) = \cos(\pi)+i \sin(\pi)
6885625907 \exp(i \pi) = -1 + i 0
3331824625 \exp(i \pi) = -1
4901237716 1
2501591100 \exp(i \pi) + 1 = 0
5136652623 E = KE + PE mechanical energy is the sum of the potential plus kinetic energies
7915321076 E, KE, PE  
3192374582 E_2, KE_2, PE_2
 
7875206161 E_2 = KE_2 + PE_2
4229092186 E, KE, PE  
1490821948 E_1, KE_1, PE_1
 
4303372136 E_1 = KE_1 + PE_1
5514556106 E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
8357234146 KE = \frac{1}{2} m v^2 kinetic energy Kinetic_energy
3658066792 KE_2, v_2
 
8082557839 KE, v  
7676652285 KE_2 = \frac{1}{2} m v_2^2
2790304597 KE_1, v_1
 
9880327189 KE, v  
4928007622 KE_1 = \frac{1}{2} m v_1^2
5733146966 KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
6554292307 t
4270680309 \frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
5781981178 x^2 - y^2 = (x+y)(x-y) difference of squares Difference_of_two_squares
5946759357 v_2, v_1
 
6675701064 x, y  
4648451961 v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
9356924046 \frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
9397152918 v = \frac{v_1 + v_2}{2} average velocity
2857430695 a = \frac{v_2 - v_1}{t}
acceleration
7735737409 \frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
4784793837 \frac{KE_2 - KE_1}{t} = m v a
5345738321 F = m a Newton's second law of motion Newton%27s_laws_of_motion#Newton's_second_law
2186083170 \frac{KE_2 - KE_1}{t} = v F
6715248283 PE = -F x potential energy Potential_energy
3852078149 PE_2, x_2
 
7388921188 PE, x  
2431507955 PE_2 = -F x_2
3412641898 PE_1, x_1
 
9937169749 PE, x  
4669290568 PE_1 = -F x_1
7734996511 PE_2 - PE_1 = -F ( x_2 - x_1 )
2016063530 t
7267155233 \frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
9337785146 v = \frac{x_2 - x_1}{t} average velocity
4872970974 \frac{PE_2 - PE_1}{t} = -F v
2081689540 t
2770069250 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
3591237106 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
1772416655 \frac{E_2 - E_1}{t} = v F - F v
1809909100 \frac{E_2 - E_1}{t} = 0
5778176146 t
3806977900 E_2 - E_1 = 0
5960438249 E_1
8558338742 E_2 = E_1
8945218208 \theta_{Brewster} + \theta_{refracted} = 90^{\circ} based on figure 34-27 on page 824 in 2001_HRW
9025853427 \theta_{Brewster}
1310571337 \theta_{refracted} = 90^{\circ} - \theta_{Brewster}
6450985774 n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 ) Law of Refraction eq 34-44 on page 819 in 2001_HRW
1779497048 \theta_{Brewster}, \theta_{refraction}  
7322344455 \theta_1, \theta_2
 
2575937347 n_1 \sin( \theta_{Brewster} ) = n_2 \sin( \theta_{refraction} )
7696214507 n_1 \sin( \theta_{Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{Brewster} )
8588429722 \sin( 90^{\circ} - x ) = \cos( x )
7375348852 \theta_{Brewster}
1512581563 x
6831637424 \sin( 90^{\circ} - \theta_{Brewster} ) = \cos( \theta_{Brewster} )
3061811650 n_1 \sin( \theta_{Brewster} ) = n_2 \cos( \theta_{Brewster} )
7857757625 n_1
9756089533 \sin( \theta_{Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{Brewster} )
5632428182 \cos( \theta_{Brewster} )
2768857871 \frac{\sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )} = \frac{n_2}{n_1}
4968680693 \tan( x ) = \frac{ \sin( x )}{\cos( x )}
7321695558 \theta_{Brewster}
9906920183 x
4501377629 \tan( \theta_{Brewster} ) = \frac{ \sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )}
3417126140 \tan( \theta_{Brewster} ) = \frac{ n_2 }{ n_1 }
5453995431 \arctan{ x }
6023986360 x
8495187962 \theta_{Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
9040079362 f
9565166889 T
5426308937 v = \frac{d}{t}
7476820482 C
1277713901 d
6946088325 v = \frac{C}{t}
6785303857 C = 2 \pi r
4816195267 C, r
 
2113447367 C_{\rm{Earth\ orbit}}, r_{\rm{Earth\ orbit}}
 
6348260313 C_{\rm{Earth\ orbit}} = 2 \pi r_{\rm{Earth\ orbit}}
3847777611 C, v, t  
6406487188 C_{\rm{Earth\ orbit}}, v_{\rm{Earth\ orbit}}, t_{\rm{Earth\ orbit}}
 
3046191961 v_{\rm{Earth\ orbit}} = \frac{C_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
3080027960 v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
7175416299 t_{\rm{Earth\ orbit}} = 1 {\rm{year}}
3219318145 \frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}
8721295221 t_{\rm{Earth\ orbit}} = 3.16 10^7 {\rm{seconds}}
4593428198 v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{3.16\ 10^7 {\rm seconds}}
3472836147 r_{\rm{Earth\ orbit}} = 1.496\ 10^8 {\rm km}
6998364753 v_{\rm{Earth\ orbit}} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
4180845508 v_{\rm{Earth\ orbit}} = 29.8 \frac{{\rm km}}{{\rm sec}}
4662369843 x' = \gamma (x - v t)
2983053062 x = \gamma (x' + v t')
3426941928 x = \gamma ( \gamma (x - v t) + v t' )
2096918413 x = \gamma ( \gamma x - \gamma v t + v t' )
7741202861 x = \gamma^2 x - \gamma^2 v t + \gamma v t'
7337056406 \gamma^2 x
4139999399 x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
3495403335 x
9031609275 x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
8014566709 \gamma^2 v t
9409776983 x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
2226340358 \gamma v
6417705759 \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = \gamma v t'
 
8730201316 \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'
first term was multiplied by \gamma/\gamma
1974334644 \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
5148266645 t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
4287102261 x^2 + y^2 + z^2 = c^2 t^2 describes a spherical wavefront
1201689765 x'^2 + y'^2 + z'^2 = c^2 t'^2
describes a spherical wavefront for an observer in a moving frame of reference
7057864873 y' = y
frame of reference is moving only along x direction
8515803375 z' = z
frame of reference is moving only along x direction
6153463771 3
 
4746576736 asdf
 
1027270623 minga
 
6101803533 ming
 
9231495607 a = b
 
8664264194 b + 2
 
9805063945 \gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2
4057686137 C \to C_{\rm{Earth\ orbit}}
2346150725 r \to r_{\rm{Earth\ orbit}}
9753878784 v \to v_{\rm{Earth\ orbit}}
8135396036 t \to t_{\rm{Earth\ orbit}}
2773628333 \theta_1 \to \theta_{Brewster}
7154592211 \theta_2 \to \theta_{\rm refracted}
3749492596 E \to E_1
1258245373 E \to E_2
4147101187 KE \to KE_1
3809726424 PE \to PE_1
6383056612 KE \to KE_2
5075406409 PE \to PE_2
5398681503 v \to v_1
9305761407 v \to v_2
4218009993 x \to x_1
4188639044 x \to x_2
8173074178 x \to v_2
1025759423 y \to v_1
1935543849 \gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2
1586866563 \left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)
3182633789 \gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
1916173354 -\gamma^2 v^2 + c^2 \gamma^2 = c^2
2076171250 -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0
5284610349 \gamma^2
2417941373 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
5787469164 1 - \gamma^2
1639827492 - c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
5669500954 v^2 \gamma^2
5763749235 -c^2 + c^2 \gamma^2 = v^2 \gamma^2
6408214498 c^2
2999795755 c^2 \gamma^2 = v^2 \gamma^2 + c^2
3412946408 v^2 \gamma^2
2542420160 c^2 \gamma^2 - v^2 \gamma^2 = c^2
7743841045 \gamma^2
7513513483 \gamma^2 (c^2 - v^2) = c^2
8571466509 c^2 - \gamma^2
7906112355 \gamma^2 = \frac{c^2}{c^2 - \gamma^2}
3060139639 1/2
 
1528310784 \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
8360117126 \gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}} not a physically valid result in this context
3366703541 a = \frac{v - v_0}{t} acceleration is the average change in speed over a duration
7083390553 t
4748157455 a t = v - v_0
6417359412 v_0
4798787814 a t + v_0 = v
3462972452 v = v_0 + a t
3411994811 v_{\rm ave} = \frac{d}{t}
6175547907 v_{\rm ave} = \frac{v + v_0}{2}
9897284307 \frac{d}{t} = \frac{v + v_0}{2}
8865085668 t
8706092970 d = \left(\frac{v + v_0}{2}\right)t
7011114072 d = \frac{(v_0 + a t) + v_0}{2} t
1265150401 d = \frac{2 v_0 + a t}{2} t
9658195023 d = v_0 t + \frac{1}{2} a t^2
5799753649 2
7215099603 v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
5144263777 v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
7939765107 v^2 = v_0^2 + 2 a d
6729698807 v_0
9759901995 v - v_0 = a t
2242144313 a
1967582749 t = \frac{v - v_0}{a}
5733721198 d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
5611024898 d = \frac{1}{2 a} (v^2 - v_0^2)
5542390646 2 a
8269198922 2 a d = v^2 - v_0^2
9070454719 v_0^2
4948763856 2 a d + v_0^2 = v^2
9645178657 a t
6457044853 v - a t = v_0
1259826355 d = (v - a t) t + \frac{1}{2} a t^2
4580545876 d = v t - a t^2 + \frac{1}{2} a t^2
6421241247 d = v t - \frac{1}{2} a t^2
4182362050 Z = |Z| \exp( i \theta ) Z \in \mathbb{C}
1928085940 Z^* = |Z| \exp( -i \theta )
9191880568 Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
3350830826 Z Z^* = |Z|^2
8396997949 I = | A + B |^2 intensity of two waves traveling opposite directions on same path
4437214608 Z
5623794884 A + B
2236639474 (A + B)(A + B)^* = |A + B|^2
1020854560 I = (A + B)(A + B)^*
6306552185 I = (A + B)(A^* + B^*)
8065128065 I = A A^* + B B^* + A B^* + B A^*
9761485403 Z
8710504862 A
4075539836 A A^* = |A|^2
6529120965 B
1511199318 Z
7107090465 B B^* = |B|^2
5125940051 I = |A|^2 + B B^* + A B^* + B A^*
1525861537 I = |A|^2 + |B|^2 + A B^* + B A^*
1742775076 Z \to B
7607271250 \theta \to \phi
4192519596 B = |B| \exp(i \phi)
2064205392 A
1894894315 Z
1357848476 A = |A| \exp(i \theta)
6555185548 A^* = |A| \exp(-i \theta)
4504256452 B^* = |B| \exp(-i \phi)
7621705408 I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
3085575328 I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
4935235303 x
2293352649 \theta - \phi
3660957533 \cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
3967985562 2
2700934933 2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
8497631728 I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
6774684564 \theta = \phi for coherent waves
2099546947 I = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )  
2719691582 |A| = |B| in a loop
3257276368 I = |A|^2 + |A|^2 + |A| |A| 2
 
8283354808 I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
8046208134 I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
1172039918 I_{\rm coherent} = 4 |A|^2
8602221482 \langle \cos(\theta - \phi) \rangle = 0 incoherent light source
6240206408 I_{\rm incoherent} = |A|^2 + |B|^2
6529793063 I_{\rm incoherent} = |A|^2 + |A|^2
3060393541 I_{\rm incoherent} = 2|A|^2
6556875579 \frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
8750500372 f
 
7543102525 g
 
8267133829 a = b
 
7968822469 c = d
 
3169580383 \vec{a} = \frac{d\vec{v}}{dt} acceleration is the change in speed over a duration
8602512487 \vec{a} = a_x \hat{x} + a_y \hat{y} decompose acceleration into two components
4158986868 a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
5349866551 \vec{v} = v_x \hat{x} + v_y \hat{y}
7729413831 a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
1819663717 a_x = \frac{d}{dt} v_x
8228733125 a_y = \frac{d}{dt} v_y
9707028061 a_x = 0
2741489181 a_y = -g
3270039798 2
8880467139 2
8750379055 0 = \frac{d}{dt} v_x
1977955751 -g = \frac{d}{dt} v_y
6672141531 dt
1702349646 -g dt = d v_y
8584698994 -g \int dt = \int d v_y
9973952056 -g t = v_y - v_{0, y}
4167526462 v_{0, y}
6572039835 -g t + v_{0, y} = v_y
7252338326 v_y = \frac{dy}{dt}
6204539227 -g t + v_{0, y} = \frac{dy}{dt}
1614343171 dt
8145337879 -g t dt + v_{0, y} dt = dy
8808860551 -g \int t dt + v_{0, y} \int dt = \int dy
2858549874 - \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
6098638221 y_0
2461349007 - \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
8717193282 dt
1166310428 0 dt = d v_x
2366691988 \int 0 dt = \int d v_x
1676472948 0 = v_x - v_{0, x}
1439089569 v_{0, x}
6134836751 v_{0, x} = v_x
8460820419 v_x = \frac{dx}{dt}
7455581657 v_{0, x} = \frac{dx}{dt}
8607458157 dt
1963253044 v_{0, x} dt = dx
3676159007 v_{0, x} \int dt = \int dx
9882526611 v_{0, x} t = x - x_0
3182907803 x_0
8486706976 v_{0, x} t + x_0 = x
1306360899 x = v_{0, x} t + x_0
7049769409 2
9341391925 \vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
6410818363 \theta
7391837535 \cos(\theta) = \frac{v_{0, x}}{v_0}
7376526845 \sin(\theta) = \frac{v_{0, y}}{v_0}
5868731041 v_0
6083821265 v_0 \cos(\theta) = v_{0, x}
5438722682 x = v_0 t \cos(\theta) + x_0
5620558729 v_0
8949329361 v_0 \sin(\theta) = v_{0, y}
1405465835 y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
9862900242 y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
6050070428 v_{0, x}
3274926090 t = \frac{x - x_0}{v_{0, x}}
7354529102 y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0
8406170337 y \to y_f
2403773761 t \to t_f
5379546684 y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
5373931751 t = t_f
9112191201 y_f = 0
8198310977 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
1650441634 y_0 = 0
define coordinate system such that initial height is at origin
1087417579 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
4829590294 t_f
2086924031 0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
6974054946 \frac{1}{2} g t_f
1191796961 \frac{1}{2} g t_f = v_0 \sin(\theta)
2510804451 2/g
4778077984 t_f = \frac{2 v_0 \sin(\theta)}{g}
3273630811 x \to x_f
6732786762 t \to t_f
3485125659 x_f = v_0 t_f \cos(\theta) + x_0
4370074654 t = t_f
2378095808 x_f = x_0 + d
4268085801 x_0 + d = v_0 t_f \cos(\theta) + x_0
8072682558 x_0
7233558441 d = v_0 t_f \cos(\theta)
2297105551 d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
7587034465 \theta
7214442790 x
2519058903 \sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
8922441655 d = \frac{v_0^2}{g} \sin(2 \theta)
5667870149 \theta
1541916015 \theta = \frac{\pi}{4}
3607070319 d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
5353282496 d = \frac{v_0^2}{g}
0000040490 a^2
0000999900 b/(2a)
0001030901 \cos(x)
0001111111 (\sin(x))^2
0001209482 2 \pi
0001304952 \hbar
0001334112 W
0001921933 2 i
0002239424 2
0002338514 \vec{p}_{2}
0002342425 m/m
0002367209 1/2
 
0002393922 x
0002424922 a
0002436656 i \hbar
0002449291 b/(2a)
0002838490 b/(2a)
0002919191 \sin(-x)
0002929944 1/2
0002940021 2 \pi
0003232242 t
0003413423 \cos(-x)
0003747849 -1
0003838111 2
0003919391 x
0003949052 -x
0003949921 \hbar
0003954314 dx
0003981813 -\sin(x)
0004089571 2x
0004264724 y
0004307451 (b/(2a))^2
0004582412 x
0004829194 2
0004831494 a
0004849392 x
0004858592 h
0004934845 x
0004948585 a
0005395034 a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
0005626421 t
0005749291 f
0005938585 \frac{-\hbar^2}{2m}
0006466214 (\sin(x))^2
0006544644 t
0006563727 x
0006644853 c/a
0006656532 e
0007471778 2(\sin(x))^2
0007563791 i
0007636749 x
0007894942 (\sin(x))^2
0008837284 T
0008842811 \cos(2x)
0009458842 \psi(x)
0009484724 \frac{n \pi}{W}x
0009485857 a^2\frac{2}{W}
0009485858 \frac{2n\pi}{W}
0009492929 v du
0009587738 \psi
0009594995 1/2
 
0009877781 y
0203024440 1 = \int_0^W a \sin(\frac{n \pi}{W} x) \psi(x)^* dx
0404050504 \lambda = \frac{v}{f}
0439492440 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\right|_0^W
0934990943 k = \frac{2 \pi}{v T}
0948572140 \int \cos(a x) dx = \frac{1}{a}\sin(a x)
1010393913 \langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*
1010393944 x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
1010923823 k W = n \pi
1020010291 0 = a \sin(k W)
1020394900 p = h/\lambda
1020394902 E = h f
1029039903 p = m v
1029039904 p^2 = m^2 v^2
1158485859 \frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
1202310110 \frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1202312210 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1203938249 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
1248277773 \cos(2x) = 1-2(\sin(x))^2
1293913110 0 = b
1293923844 \lambda = v T
1314464131 \vec{\nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
1314864131 \vec{\nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
1395858355 x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
1492811142 f = x - d  
1492842000 \nabla \vec{x} = f(y)  
1636453295 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = - \nabla^2 \vec{E}
1638282134 \vec{p}_{before} = \vec{p}_{after}
1648958381 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{\nabla}\psi(\vec{r},t)\right)
1857710291 0 = a \sin(n \pi)
1858578388 \nabla^2 E(\vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E(\vec{r})\exp(i \omega t)
1858772113 k = \frac{n \pi}{W}
1934748140 \int |\psi(x)|^2 dx = 1
2029293929 \nabla^2 E(\vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r})\exp(i \omega t)
2103023049 \sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right)
2113211456 f = 1/T
2123139121 -\exp(-i x) = -\cos(x)+i \sin(x)
2131616531 T f = 1
2258485859 {\cal H} \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
2394240499 x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
2394853829 \exp(-i x) = \cos(-x)+i \sin(-x)
2394935831 ( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
2394935835 \left(\langle\psi| \hat{A} |\psi\rangle \right)^+ = \left(\langle a \rangle\right)^+
2395958385 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{-p^2}{\hbar} \psi(\vec{r},t)
2404934990 \langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
2494533900 \langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
2569154141 \vec{\nabla} \times \frac{\partial}{\partial t}\vec{H} = \epsilon_0 \frac{\partial^2 }{\partial t^2}\vec{E}  
2648958382 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p}\psi(\vec{r},t)\right)
2848934890 \langle a \rangle^* = \langle a \rangle
2944838499 \psi(x) = a \sin(\frac{n \pi}{W} x)
3121234211 \frac{k}{2\pi} = \lambda
3121234212 p = \frac{h k}{2\pi}
3121513111 k = \frac{2 \pi}{\lambda}
3131111133 T = 1 / f
3131211131 \omega = 2 \pi f
3132131132 \omega = \frac{2\pi}{T}
3147472131 \frac{\omega}{2 \pi} = f
3285732911 (\cos(x))^2 = 1-(\sin(x))^2
3485475729 \nabla^2 E(\vec{r}) = - \frac{\omega^2}{c^2} E(\vec{r})
3585845894 \langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
3829492824 \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right) = \cos(x)
3924948349 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
3942849294 \exp(i x)-\exp(-i x) = 2 i \sin(x)
3943939590 x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
3947269979 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
3948571256 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi(\vec{r},t)
3948572230 \vec{\nabla}\psi(\vec{r},t) = \psi_0 \vec{\nabla}\exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)  
3948574224 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\vec{k}\cdot\vec{r} - \omega t \right) \right)
3948574226 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
3948574228 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3948574230 \psi(\vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
3948574233 \frac{\partial}{\partial t}\psi(\vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3948574235 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)  
3951205425 \vec{p}_{after} = \vec{p}_{1}
4147472132 E = \frac{h \omega}{2 \pi}
4298359835 E = \frac{1}{2}m v^2
4298359845 E = \frac{1}{2m}m^2 v^2
4298359851 E = \frac{p^2}{2m}
4341171256 i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{p^2}{2 m} \psi(\vec{r},t)
4348571256 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi(\vec{r},t)
4394958389 \vec{\nabla}\cdot \left(\vec{\nabla}\psi(\vec{r},t)\right) = \frac{i}{\hbar} \vec{\nabla}\cdot\left( \vec{p} \psi(\vec{r},t)\right)
4585828572 \epsilon_0 \mu_0 = \frac{1}{c^2}
4585932229 \cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
4598294821 \exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
4638429483 \exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
4742644828 \exp(i x)+\exp(-i x) = 2 \cos(x)
4827492911 \cos(2x)+(\sin(x))^2 = 1-(\sin(x))^2
4838429483 \exp(2 i x) = \cos(2 x)+i \sin(2 x)
4843995999 \frac{1}{2 i}\left(\exp(i x)-\exp(-i x)\right) = \sin(x)
4857472413 1 = \int \psi(x)\psi(x)^* dx
4857475848 \frac{1}{a^2} = \frac{W}{2}
4858429483 \exp(i x)\exp(i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))  
4923339482 i x = \log(y)
4928239482 \log(y) = i x
4928923942 a = b  
4929218492 a+b = c  
4929482992 b = 2
 
4938429482 \exp(-i x) = \cos(x)+i \sin(-x)
4938429483 \exp(i x) = \cos(x)+i \sin(x)
4938429484 \exp(-i x) = \cos(x)-i \sin(x)
4943571230 \vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
4948934890 \langle \psi| \hat{A} |\psi\rangle = \langle a \rangle^*
4949359835 \langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
4954839242 \cos(2x)+i\sin(2x) = (\cos(x)+i \sin(x))(\cos(x)+i \sin(x))  
4978429483 \exp(-i x) = \cos(-x)+i \sin(-x)  
4984892984 a+2 = c
 
4985825552 \nabla^2 E(\vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E(\vec{r})\exp(i \omega t)
5530148480 \vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
5727578862 \frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
5832984291 (\sin(x))^2 + (\cos(x))^2 = 1
5857434758 \int a dx = a x
5868688585 \frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = \frac{p^2}{2m} \psi(\vec{r},t)
5900595848 k = \frac{\omega}{v}
5928285821 x^2 + 2x(b/(2a)) + (b/(2a))^2 = (x+(b/(2a)))^2
5928292841 x^2 + (b/a)x + (b/(2a))^2 = -c/a + (b/(2a))^2
5938459282 x^2 + (b/a)x = -c/a
5958392859 x^2 + (b/a)x+(c/a) = 0
5959282914 x^2 + x(b/a) + (b/(2a))^2 = (x+(b/(2a)))^2
5982958248 x = -\sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
5982958249 x+(b/(2a)) = -\sqrt{(b/(2a))^2 - (c/a)}
5985371230 \vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi(\vec{r},t)
6742123016 \vec{p}_{electron}\cdot\vec{p}_{electron} = (\vec{p}_{1}\cdot\vec{p}_{1})+(\vec{p}_{2}\cdot\vec{p}_{2})-2(\vec{p}_{1}\cdot\vec{p}_{2})
7466829492 \vec{\nabla} \cdot \vec{E} = 0
7564894985 \int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
7572664728 \cos(2x)+2(\sin(x))^2 = 1
7575738420 \left(\sin\left(\frac{n \pi}{W}x\right)\right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
7575859295 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859300 \epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{\nabla} \times \vec{E} )_k = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859302 \epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859304 \epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
7575859306 \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859308 \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859310 \hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859312 \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7917051060 \vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
8139187332 \vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
8257621077 \vec{p}_{before} = \vec{p}_{1}
8311458118 \vec{p}_{after} = \vec{p}_{2}+\vec{p}_{electron}
8399484849 \langle x^2 -2x\langle x \rangle+\langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
8484544728 -a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
8485757728 a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
8485867742 \frac{2}{W} = a^2
8489593958 d(u v) = u dv + v du
8489593960 d(u v) - v du = u dv
8489593962 u dv = d(u v) - v du
8489593964 \int u dv = u v - \int v du
8494839423 \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
8572657110 1 = \int |\psi(x)|^2 dx
8572852424 \vec{E} = E(\vec{r},t)
8575746378 \int \frac{1}{2} dx = \frac{1}{2} x
8575748999 \frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x)\right) = -k^2 \left(a \sin(kx) + b \cos(kx)\right)
8576785890 1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
8577275751 0 = a \sin(0) + b\cos(0)
8582885111 \psi(x) = a \sin(kx) + b \cos(kx)
8582954722 x^2 + 2xh + h^2 = (x+h)^2
8849289982 \psi(x)^* = a \sin(\frac{n \pi}{W} x)
8889444440 1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right)\right)^2 dx
9059289981 \psi(x) = a \sin(k x)
9285928292 ax^2 + bx + c = 0
9291999979 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t}
9294858532 \hat{A}^+ = \hat{A}
9385938295 (x+(b/(2a)))^2 = -(c/a) + (b/(2a))^2
9393939991 \psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9393939992 \psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9394939493 \nabla^2 E(\vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r},t)
9429829482 \frac{d}{dx} y = -\sin(x) + i\cos(x)
9482113948 \frac{dy}{y} = i dx
9482438243 (\cos(x))^2 = \cos(2x)+(\sin(x))^2
9482923849 \exp(i x) = y
9482928242 \cos(2x) = (\cos(x))^2 - (\sin(x))^2
9482928243 \cos(2x)+(\sin(x))^2 = (\cos(x))^2
9482943948 \log(y) = i dx
9482948292 b = c
 
9482984922 \frac{d}{dx} y = (i\sin(x) + \cos(x)) i
9483928192 \cos(2x)+i\sin(2x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
9485384858 \nabla^2 E(\vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E(\vec{r})\exp(i \omega t)
9485747245 \sqrt{\frac{2}{W}} = a
9485747246 -\sqrt{\frac{2}{W}} = a
9492920340 y = \cos(x)+i \sin(x)
9494829190 k  
9495857278 \psi(x=W) = 0
9499428242 E(\vec{r},t) = E(\vec{r})\exp(i \omega t)
9499959299 a + k = b + k  
9582958293 x = \sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
9582958294 x+(b/(2a)) = \sqrt{(b/(2a))^2 - (c/a)}
9584821911 c + d = a  
9585727710 \psi(x=0) = 0
9596004948 x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
9848292229 dy = y i dx
9848294829 \frac{d}{dx} y = y i
9858028950 \frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
9889984281 2(\sin(x))^2 = 1-\cos(2x)
9919999981 \rho = 0
9958485859 \frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
9988949211 (\sin(x))^2 = \frac{1-\cos(2x)}{2}
9991999979 \vec{\nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
9999998870 \frac{\vec{p}}{\hbar} = \vec{k}
9999999870 \frac{p}{\hbar} = k
9999999950 \beta = 1/(k_{Boltzmann}T)  
9999999951 \langle x | k \rangle = \frac{\exp(i k x)}{\sqrt{2\pi}}  
9999999952 f(x) \delta(x-a) = f(a)
 
9999999953 \int_{-\infty}^{\infty} \delta(x) dx = 1
 
9999999954 c = 1/\sqrt{\epsilon_0 \mu_0}  
9999999955 \vec{E} = -\vec{\nabla} \Psi
 
9999999956 \vec{F} = \frac{\partial}{\partial t}\vec{p}  
9999999957 \vec{F} = -\vec{\nabla} V
 
9999999960 \hbar = h/(2 \pi)
9999999961 \frac{E}{\hbar} = \omega
9999999962 p = \hbar k
9999999963 \lambda = h/p
 
9999999964 \omega = c k  
9999999965 E = \omega \hbar
9999999966 \vec{L} = \vec{r}\times\vec{p}
 
9999999967 \vec{S} = \frac{1}{\mu_0} \vec{E}\times \vec{B}
 
9999999968 x = \frac{-b-\sqrt{b^2-4ac}}{2a}
9999999969 x = \frac{-b+\sqrt{b^2-4ac}}{2a}
9999999970 \eta_1 \sin\theta_1 = \eta_2 \sin\theta_2
 
9999999971 {\cal H} = \frac{p^2}{2m}+V
 
9999999972 {\cal H} |\psi\rangle = E |\psi\rangle
 
9999999973 \left( \Delta A \right)^2 = \langle A^2 \rangle - \langle A \rangle^2
 
9999999974 \langle \psi| \hat{A} |\psi\rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx
 
9999999975 \langle \psi| \hat{A} |\psi\rangle = \langle a \rangle
9999999976 \hat{p} = -i \hbar \frac{\partial }{\partial x}  
9999999977 [\hat{x},\hat{p}] = i \hbar  
9999999978 \vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}
 
9999999979 \vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}  
9999999980 \vec{\nabla} \cdot \vec{B} = 0
 
9999999981 \vec{\nabla} \cdot \vec{E} = \rho/\epsilon_0
9999999982 V = I R + Q/C + L \frac{\partial I}{\partial t}  
9999999983 C = V A/d
 
9999999984 Q = C V
 
9999999985 V = I R
 
9999999986 \left[cos⁡θsin⁡θ−sin⁡θcos⁡θ\right]\left[AxAy\right] = \left[Ax′Ay′\right]
 
9999999987 1 atmosphere = 101.325 Pascal
 
9999999988 1 atmosphere = 14.7 pounds/(inch^2)
 
9999999989 mass_{electron} = 511000 electronVolts/(q^2)
 
9999999990 Tesla = 10000*Gauss
 
9999999991 Pascal = Newton / (meter^2)
 
9999999992 Tesla = Newton / (Ampere*meter)
 
9999999993 Farad = Coulumb / Volt
 
9999999995 Volt = Joule / Coulumb
 
9999999996 Coulomb = Ampere / second
 
9999999997 Watt = Joule / second
 
9999999998 Joule = Newton*meter
 
9999999999 Newton = kilogram*meter/(second^2)
 
4961662865 x
9110536742 2 x
8483686863 \sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x)\right)
3470587782 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
8642992037 2
9894826550 2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \left(\exp(i x)+\exp(-i x)\right)
4257214632 \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)  
8699789241 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
9180861128 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
2405307372 \sin(2 x) = 2 \sin(x) \cos(x)
3268645065 x
9350663581 \pi
8332931442 \exp(i \pi) = \cos(\pi)+i \sin(\pi)
6885625907 \exp(i \pi) = -1 + i 0
3331824625 \exp(i \pi) = -1
4901237716 1
2501591100 \exp(i \pi) + 1 = 0
5136652623 E = KE + PE mechanical energy is the sum of the potential plus kinetic energies
7915321076 E, KE, PE  
3192374582 E_2, KE_2, PE_2
 
7875206161 E_2 = KE_2 + PE_2
4229092186 E, KE, PE  
1490821948 E_1, KE_1, PE_1
 
4303372136 E_1 = KE_1 + PE_1
5514556106 E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
8357234146 KE = \frac{1}{2} m v^2 kinetic energy Kinetic_energy
3658066792 KE_2, v_2
 
8082557839 KE, v  
7676652285 KE_2 = \frac{1}{2} m v_2^2
2790304597 KE_1, v_1
 
9880327189 KE, v  
4928007622 KE_1 = \frac{1}{2} m v_1^2
5733146966 KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
6554292307 t
4270680309 \frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
5781981178 x^2 - y^2 = (x+y)(x-y) difference of squares Difference_of_two_squares
5946759357 v_2, v_1
 
6675701064 x, y  
4648451961 v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
9356924046 \frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
9397152918 v = \frac{v_1 + v_2}{2} average velocity
2857430695 a = \frac{v_2 - v_1}{t}
acceleration
7735737409 \frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
4784793837 \frac{KE_2 - KE_1}{t} = m v a
5345738321 F = m a Newton's second law of motion Newton%27s_laws_of_motion#Newton's_second_law
2186083170 \frac{KE_2 - KE_1}{t} = v F
6715248283 PE = -F x potential energy Potential_energy
3852078149 PE_2, x_2
 
7388921188 PE, x  
2431507955 PE_2 = -F x_2
3412641898 PE_1, x_1
 
9937169749 PE, x  
4669290568 PE_1 = -F x_1
7734996511 PE_2 - PE_1 = -F ( x_2 - x_1 )
2016063530 t
7267155233 \frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
9337785146 v = \frac{x_2 - x_1}{t} average velocity
4872970974 \frac{PE_2 - PE_1}{t} = -F v
2081689540 t
2770069250 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
3591237106 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
1772416655 \frac{E_2 - E_1}{t} = v F - F v
1809909100 \frac{E_2 - E_1}{t} = 0
5778176146 t
3806977900 E_2 - E_1 = 0
5960438249 E_1
8558338742 E_2 = E_1
8945218208 \theta_{Brewster} + \theta_{refracted} = 90^{\circ} based on figure 34-27 on page 824 in 2001_HRW
9025853427 \theta_{Brewster}
1310571337 \theta_{refracted} = 90^{\circ} - \theta_{Brewster}
6450985774 n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 ) Law of Refraction eq 34-44 on page 819 in 2001_HRW
1779497048 \theta_{Brewster}, \theta_{refraction}  
7322344455 \theta_1, \theta_2
 
2575937347 n_1 \sin( \theta_{Brewster} ) = n_2 \sin( \theta_{refraction} )
7696214507 n_1 \sin( \theta_{Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{Brewster} )
8588429722 \sin( 90^{\circ} - x ) = \cos( x )
7375348852 \theta_{Brewster}
1512581563 x
6831637424 \sin( 90^{\circ} - \theta_{Brewster} ) = \cos( \theta_{Brewster} )
3061811650 n_1 \sin( \theta_{Brewster} ) = n_2 \cos( \theta_{Brewster} )
7857757625 n_1
9756089533 \sin( \theta_{Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{Brewster} )
5632428182 \cos( \theta_{Brewster} )
2768857871 \frac{\sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )} = \frac{n_2}{n_1}
4968680693 \tan( x ) = \frac{ \sin( x )}{\cos( x )}
7321695558 \theta_{Brewster}
9906920183 x
4501377629 \tan( \theta_{Brewster} ) = \frac{ \sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )}
3417126140 \tan( \theta_{Brewster} ) = \frac{ n_2 }{ n_1 }
5453995431 \arctan{ x }
6023986360 x
8495187962 \theta_{Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
9040079362 f
9565166889 T
5426308937 v = \frac{d}{t}
7476820482 C
1277713901 d
6946088325 v = \frac{C}{t}
6785303857 C = 2 \pi r
4816195267 C, r
 
2113447367 C_{\rm{Earth\ orbit}}, r_{\rm{Earth\ orbit}}
 
6348260313 C_{\rm{Earth\ orbit}} = 2 \pi r_{\rm{Earth\ orbit}}
3847777611 C, v, t  
6406487188 C_{\rm{Earth\ orbit}}, v_{\rm{Earth\ orbit}}, t_{\rm{Earth\ orbit}}
 
3046191961 v_{\rm{Earth\ orbit}} = \frac{C_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
3080027960 v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
7175416299 t_{\rm{Earth\ orbit}} = 1 {\rm{year}}
3219318145 \frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}
8721295221 t_{\rm{Earth\ orbit}} = 3.16 10^7 {\rm{seconds}}
4593428198 v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{3.16\ 10^7 {\rm seconds}}
3472836147 r_{\rm{Earth\ orbit}} = 1.496\ 10^8 {\rm km}
6998364753 v_{\rm{Earth\ orbit}} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
4180845508 v_{\rm{Earth\ orbit}} = 29.8 \frac{{\rm km}}{{\rm sec}}
4662369843 x' = \gamma (x - v t)
2983053062 x = \gamma (x' + v t')
3426941928 x = \gamma ( \gamma (x - v t) + v t' )
2096918413 x = \gamma ( \gamma x - \gamma v t + v t' )
7741202861 x = \gamma^2 x - \gamma^2 v t + \gamma v t'
7337056406 \gamma^2 x
4139999399 x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
3495403335 x
9031609275 x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
8014566709 \gamma^2 v t
9409776983 x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
2226340358 \gamma v
6417705759 \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = \gamma v t'
 
8730201316 \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'
first term was multiplied by \gamma/\gamma
1974334644 \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
5148266645 t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
4287102261 x^2 + y^2 + z^2 = c^2 t^2 describes a spherical wavefront
1201689765 x'^2 + y'^2 + z'^2 = c^2 t'^2
describes a spherical wavefront for an observer in a moving frame of reference
7057864873 y' = y
frame of reference is moving only along x direction
8515803375 z' = z
frame of reference is moving only along x direction
6153463771 3
 
4746576736 asdf
 
1027270623 minga
 
6101803533 ming
 
9231495607 a = b
 
8664264194 b + 2
 
9805063945 \gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2
4057686137 C \to C_{\rm{Earth\ orbit}}
2346150725 r \to r_{\rm{Earth\ orbit}}
9753878784 v \to v_{\rm{Earth\ orbit}}
8135396036 t \to t_{\rm{Earth\ orbit}}
2773628333 \theta_1 \to \theta_{Brewster}
7154592211 \theta_2 \to \theta_{\rm refracted}
3749492596 E \to E_1
1258245373 E \to E_2
4147101187 KE \to KE_1
3809726424 PE \to PE_1
6383056612 KE \to KE_2
5075406409 PE \to PE_2
5398681503 v \to v_1
9305761407 v \to v_2
4218009993 x \to x_1
4188639044 x \to x_2
8173074178 x \to v_2
1025759423 y \to v_1
1935543849 \gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2
1586866563 \left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)
3182633789 \gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
1916173354 -\gamma^2 v^2 + c^2 \gamma^2 = c^2
2076171250 -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0
5284610349 \gamma^2
2417941373 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
5787469164 1 - \gamma^2
1639827492 - c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
5669500954 v^2 \gamma^2
5763749235 -c^2 + c^2 \gamma^2 = v^2 \gamma^2
6408214498 c^2
2999795755 c^2 \gamma^2 = v^2 \gamma^2 + c^2
3412946408 v^2 \gamma^2
2542420160 c^2 \gamma^2 - v^2 \gamma^2 = c^2
7743841045 \gamma^2
7513513483 \gamma^2 (c^2 - v^2) = c^2
8571466509 c^2 - \gamma^2
7906112355 \gamma^2 = \frac{c^2}{c^2 - \gamma^2}
3060139639 1/2
 
1528310784 \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
8360117126 \gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}} not a physically valid result in this context
3366703541 a = \frac{v - v_0}{t} acceleration is the average change in speed over a duration
7083390553 t
4748157455 a t = v - v_0
6417359412 v_0
4798787814 a t + v_0 = v
3462972452 v = v_0 + a t
3411994811 v_{\rm ave} = \frac{d}{t}
6175547907 v_{\rm ave} = \frac{v + v_0}{2}
9897284307 \frac{d}{t} = \frac{v + v_0}{2}
8865085668 t
8706092970 d = \left(\frac{v + v_0}{2}\right)t
7011114072 d = \frac{(v_0 + a t) + v_0}{2} t
1265150401 d = \frac{2 v_0 + a t}{2} t
9658195023 d = v_0 t + \frac{1}{2} a t^2
5799753649 2
7215099603 v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
5144263777 v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
7939765107 v^2 = v_0^2 + 2 a d
6729698807 v_0
9759901995 v - v_0 = a t
2242144313 a
1967582749 t = \frac{v - v_0}{a}
5733721198 d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
5611024898 d = \frac{1}{2 a} (v^2 - v_0^2)
5542390646 2 a
8269198922 2 a d = v^2 - v_0^2
9070454719 v_0^2
4948763856 2 a d + v_0^2 = v^2
9645178657 a t
6457044853 v - a t = v_0
1259826355 d = (v - a t) t + \frac{1}{2} a t^2
4580545876 d = v t - a t^2 + \frac{1}{2} a t^2
6421241247 d = v t - \frac{1}{2} a t^2
4182362050 Z = |Z| \exp( i \theta ) Z \in \mathbb{C}
1928085940 Z^* = |Z| \exp( -i \theta )
9191880568 Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
3350830826 Z Z^* = |Z|^2
8396997949 I = | A + B |^2 intensity of two waves traveling opposite directions on same path
4437214608 Z
5623794884 A + B
2236639474 (A + B)(A + B)^* = |A + B|^2
1020854560 I = (A + B)(A + B)^*
6306552185 I = (A + B)(A^* + B^*)
8065128065 I = A A^* + B B^* + A B^* + B A^*
9761485403 Z
8710504862 A
4075539836 A A^* = |A|^2
6529120965 B
1511199318 Z
7107090465 B B^* = |B|^2
5125940051 I = |A|^2 + B B^* + A B^* + B A^*
1525861537 I = |A|^2 + |B|^2 + A B^* + B A^*
1742775076 Z \to B
7607271250 \theta \to \phi
4192519596 B = |B| \exp(i \phi)
2064205392 A
1894894315 Z
1357848476 A = |A| \exp(i \theta)
6555185548 A^* = |A| \exp(-i \theta)
4504256452 B^* = |B| \exp(-i \phi)
7621705408 I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
3085575328 I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
4935235303 x
2293352649 \theta - \phi
3660957533 \cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
3967985562 2
2700934933 2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
8497631728 I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
6774684564 \theta = \phi for coherent waves
2099546947 I = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )  
2719691582 |A| = |B| in a loop
3257276368 I = |A|^2 + |A|^2 + |A| |A| 2
 
8283354808 I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
8046208134 I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
1172039918 I_{\rm coherent} = 4 |A|^2
8602221482 \langle \cos(\theta - \phi) \rangle = 0 incoherent light source
6240206408 I_{\rm incoherent} = |A|^2 + |B|^2
6529793063 I_{\rm incoherent} = |A|^2 + |A|^2
3060393541 I_{\rm incoherent} = 2|A|^2
6556875579 \frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
8750500372 f
 
7543102525 g
 
8267133829 a = b
 
7968822469 c = d
 
3169580383 \vec{a} = \frac{d\vec{v}}{dt} acceleration is the change in speed over a duration
8602512487 \vec{a} = a_x \hat{x} + a_y \hat{y} decompose acceleration into two components
4158986868 a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
5349866551 \vec{v} = v_x \hat{x} + v_y \hat{y}
7729413831 a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
1819663717 a_x = \frac{d}{dt} v_x
8228733125 a_y = \frac{d}{dt} v_y
9707028061 a_x = 0
2741489181 a_y = -g
3270039798 2
8880467139 2
8750379055 0 = \frac{d}{dt} v_x
1977955751 -g = \frac{d}{dt} v_y
6672141531 dt
1702349646 -g dt = d v_y
8584698994 -g \int dt = \int d v_y
9973952056 -g t = v_y - v_{0, y}
4167526462 v_{0, y}
6572039835 -g t + v_{0, y} = v_y
7252338326 v_y = \frac{dy}{dt}
6204539227 -g t + v_{0, y} = \frac{dy}{dt}
1614343171 dt
8145337879 -g t dt + v_{0, y} dt = dy
8808860551 -g \int t dt + v_{0, y} \int dt = \int dy
2858549874 - \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
6098638221 y_0
2461349007 - \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
8717193282 dt
1166310428 0 dt = d v_x
2366691988 \int 0 dt = \int d v_x
1676472948 0 = v_x - v_{0, x}
1439089569 v_{0, x}
6134836751 v_{0, x} = v_x
8460820419 v_x = \frac{dx}{dt}
7455581657 v_{0, x} = \frac{dx}{dt}
8607458157 dt
1963253044 v_{0, x} dt = dx
3676159007 v_{0, x} \int dt = \int dx
9882526611 v_{0, x} t = x - x_0
3182907803 x_0
8486706976 v_{0, x} t + x_0 = x
1306360899 x = v_{0, x} t + x_0
7049769409 2
9341391925 \vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
6410818363 \theta
7391837535 \cos(\theta) = \frac{v_{0, x}}{v_0}
7376526845 \sin(\theta) = \frac{v_{0, y}}{v_0}
5868731041 v_0
6083821265 v_0 \cos(\theta) = v_{0, x}
5438722682 x = v_0 t \cos(\theta) + x_0
5620558729 v_0
8949329361 v_0 \sin(\theta) = v_{0, y}
1405465835 y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
9862900242 y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
6050070428 v_{0, x}
3274926090 t = \frac{x - x_0}{v_{0, x}}
7354529102 y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0
8406170337 y \to y_f
2403773761 t \to t_f
5379546684 y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
5373931751 t = t_f
9112191201 y_f = 0
8198310977 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
1650441634 y_0 = 0
define coordinate system such that initial height is at origin
1087417579 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
4829590294 t_f
2086924031 0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
6974054946 \frac{1}{2} g t_f
1191796961 \frac{1}{2} g t_f = v_0 \sin(\theta)
2510804451 2/g
4778077984 t_f = \frac{2 v_0 \sin(\theta)}{g}
3273630811 x \to x_f
6732786762 t \to t_f
3485125659 x_f = v_0 t_f \cos(\theta) + x_0
4370074654 t = t_f
2378095808 x_f = x_0 + d
4268085801 x_0 + d = v_0 t_f \cos(\theta) + x_0
8072682558 x_0
7233558441 d = v_0 t_f \cos(\theta)
2297105551 d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
7587034465 \theta
7214442790 x
2519058903 \sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
8922441655 d = \frac{v_0^2}{g} \sin(2 \theta)
5667870149 \theta
1541916015 \theta = \frac{\pi}{4}
3607070319 d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
5353282496 d = \frac{v_0^2}{g}
0000040490 a^2
0000999900 b/(2a)
0001030901 \cos(x)
0001111111 (\sin(x))^2
0001209482 2 \pi
0001304952 \hbar
0001334112 W
0001921933 2 i
0002239424 2
0002338514 \vec{p}_{2}
0002342425 m/m
0002367209 1/2
 
0002393922 x
0002424922 a
0002436656 i \hbar
0002449291 b/(2a)
0002838490 b/(2a)
0002919191 \sin(-x)
0002929944 1/2
0002940021 2 \pi
0003232242 t
0003413423 \cos(-x)
0003747849 -1
0003838111 2
0003919391 x
0003949052 -x
0003949921 \hbar
0003954314 dx
0003981813 -\sin(x)
0004089571 2x
0004264724 y
0004307451 (b/(2a))^2
0004582412 x
0004829194 2
0004831494 a
0004849392 x
0004858592 h
0004934845 x
0004948585 a
0005395034 a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
0005626421 t
0005749291 f
0005938585 \frac{-\hbar^2}{2m}
0006466214 (\sin(x))^2
0006544644 t
0006563727 x
0006644853 c/a
0006656532 e
0007471778 2(\sin(x))^2
0007563791 i
0007636749 x
0007894942 (\sin(x))^2
0008837284 T
0008842811 \cos(2x)
0009458842 \psi(x)
0009484724 \frac{n \pi}{W}x
0009485857 a^2\frac{2}{W}
0009485858 \frac{2n\pi}{W}
0009492929 v du
0009587738 \psi
0009594995 1/2
 
0009877781 y
0203024440 1 = \int_0^W a \sin(\frac{n \pi}{W} x) \psi(x)^* dx
0404050504 \lambda = \frac{v}{f}
0439492440 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\right|_0^W
0934990943 k = \frac{2 \pi}{v T}
0948572140 \int \cos(a x) dx = \frac{1}{a}\sin(a x)
1010393913 \langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*
1010393944 x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
1010923823 k W = n \pi
1020010291 0 = a \sin(k W)
1020394900 p = h/\lambda
1020394902 E = h f
1029039903 p = m v
1029039904 p^2 = m^2 v^2
1158485859 \frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
1202310110 \frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1202312210 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1203938249 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
1248277773 \cos(2x) = 1-2(\sin(x))^2
1293913110 0 = b
1293923844 \lambda = v T
1314464131 \vec{\nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
1314864131 \vec{\nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
1395858355 x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
1492811142 f = x - d  
1492842000 \nabla \vec{x} = f(y)  
1636453295 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = - \nabla^2 \vec{E}
1638282134 \vec{p}_{before} = \vec{p}_{after}
1648958381 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{\nabla}\psi(\vec{r},t)\right)
1857710291 0 = a \sin(n \pi)
1858578388 \nabla^2 E(\vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E(\vec{r})\exp(i \omega t)
1858772113 k = \frac{n \pi}{W}
1934748140 \int |\psi(x)|^2 dx = 1
2029293929 \nabla^2 E(\vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r})\exp(i \omega t)
2103023049 \sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right)
2113211456 f = 1/T
2123139121 -\exp(-i x) = -\cos(x)+i \sin(x)
2131616531 T f = 1
2258485859 {\cal H} \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
2394240499 x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
2394853829 \exp(-i x) = \cos(-x)+i \sin(-x)
2394935831 ( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
2394935835 \left(\langle\psi| \hat{A} |\psi\rangle \right)^+ = \left(\langle a \rangle\right)^+
2395958385 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{-p^2}{\hbar} \psi(\vec{r},t)
2404934990 \langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
2494533900 \langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
2569154141 \vec{\nabla} \times \frac{\partial}{\partial t}\vec{H} = \epsilon_0 \frac{\partial^2 }{\partial t^2}\vec{E}  
2648958382 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p}\psi(\vec{r},t)\right)
2848934890 \langle a \rangle^* = \langle a \rangle
2944838499 \psi(x) = a \sin(\frac{n \pi}{W} x)
3121234211 \frac{k}{2\pi} = \lambda
3121234212 p = \frac{h k}{2\pi}
3121513111 k = \frac{2 \pi}{\lambda}
3131111133 T = 1 / f
3131211131 \omega = 2 \pi f
3132131132 \omega = \frac{2\pi}{T}
3147472131 \frac{\omega}{2 \pi} = f
3285732911 (\cos(x))^2 = 1-(\sin(x))^2
3485475729 \nabla^2 E(\vec{r}) = - \frac{\omega^2}{c^2} E(\vec{r})
3585845894 \langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
3829492824 \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right) = \cos(x)
3924948349 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
3942849294 \exp(i x)-\exp(-i x) = 2 i \sin(x)
3943939590 x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
3947269979 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
3948571256 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi(\vec{r},t)
3948572230 \vec{\nabla}\psi(\vec{r},t) = \psi_0 \vec{\nabla}\exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)  
3948574224 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\vec{k}\cdot\vec{r} - \omega t \right) \right)
3948574226 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
3948574228 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3948574230 \psi(\vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
3948574233 \frac{\partial}{\partial t}\psi(\vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3948574235 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)  
3951205425 \vec{p}_{after} = \vec{p}_{1}
4147472132 E = \frac{h \omega}{2 \pi}
4298359835 E = \frac{1}{2}m v^2
4298359845 E = \frac{1}{2m}m^2 v^2
4298359851 E = \frac{p^2}{2m}
4341171256 i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{p^2}{2 m} \psi(\vec{r},t)
4348571256 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi(\vec{r},t)
4394958389 \vec{\nabla}\cdot \left(\vec{\nabla}\psi(\vec{r},t)\right) = \frac{i}{\hbar} \vec{\nabla}\cdot\left( \vec{p} \psi(\vec{r},t)\right)
4585828572 \epsilon_0 \mu_0 = \frac{1}{c^2}
4585932229 \cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
4598294821 \exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
4638429483 \exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
4742644828 \exp(i x)+\exp(-i x) = 2 \cos(x)
4827492911 \cos(2x)+(\sin(x))^2 = 1-(\sin(x))^2
4838429483 \exp(2 i x) = \cos(2 x)+i \sin(2 x)
4843995999 \frac{1}{2 i}\left(\exp(i x)-\exp(-i x)\right) = \sin(x)
4857472413 1 = \int \psi(x)\psi(x)^* dx
4857475848 \frac{1}{a^2} = \frac{W}{2}
4858429483 \exp(i x)\exp(i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))  
4923339482 i x = \log(y)
4928239482 \log(y) = i x
4928923942 a = b  
4929218492 a+b = c  
4929482992 b = 2
 
4938429482 \exp(-i x) = \cos(x)+i \sin(-x)
4938429483 \exp(i x) = \cos(x)+i \sin(x)
4938429484 \exp(-i x) = \cos(x)-i \sin(x)
4943571230 \vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
4948934890 \langle \psi| \hat{A} |\psi\rangle = \langle a \rangle^*
4949359835 \langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
4954839242 \cos(2x)+i\sin(2x) = (\cos(x)+i \sin(x))(\cos(x)+i \sin(x))  
4978429483 \exp(-i x) = \cos(-x)+i \sin(-x)  
4984892984 a+2 = c
 
4985825552 \nabla^2 E(\vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E(\vec{r})\exp(i \omega t)
5530148480 \vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
5727578862 \frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
5832984291 (\sin(x))^2 + (\cos(x))^2 = 1
5857434758 \int a dx = a x
5868688585 \frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = \frac{p^2}{2m} \psi(\vec{r},t)
5900595848 k = \frac{\omega}{v}
5928285821 x^2 + 2x(b/(2a)) + (b/(2a))^2 = (x+(b/(2a)))^2
5928292841 x^2 + (b/a)x + (b/(2a))^2 = -c/a + (b/(2a))^2
5938459282 x^2 + (b/a)x = -c/a
5958392859 x^2 + (b/a)x+(c/a) = 0
5959282914 x^2 + x(b/a) + (b/(2a))^2 = (x+(b/(2a)))^2
5982958248 x = -\sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
5982958249 x+(b/(2a)) = -\sqrt{(b/(2a))^2 - (c/a)}
5985371230 \vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi(\vec{r},t)
6742123016 \vec{p}_{electron}\cdot\vec{p}_{electron} = (\vec{p}_{1}\cdot\vec{p}_{1})+(\vec{p}_{2}\cdot\vec{p}_{2})-2(\vec{p}_{1}\cdot\vec{p}_{2})
7466829492 \vec{\nabla} \cdot \vec{E} = 0
7564894985 \int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
7572664728 \cos(2x)+2(\sin(x))^2 = 1
7575738420 \left(\sin\left(\frac{n \pi}{W}x\right)\right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
7575859295 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859300 \epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{\nabla} \times \vec{E} )_k = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859302 \epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859304 \epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
7575859306 \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859308 \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859310 \hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859312 \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7917051060 \vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
8139187332 \vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
8257621077 \vec{p}_{before} = \vec{p}_{1}
8311458118 \vec{p}_{after} = \vec{p}_{2}+\vec{p}_{electron}
8399484849 \langle x^2 -2x\langle x \rangle+\langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
8484544728 -a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
8485757728 a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
8485867742 \frac{2}{W} = a^2
8489593958 d(u v) = u dv + v du
8489593960 d(u v) - v du = u dv
8489593962 u dv = d(u v) - v du
8489593964 \int u dv = u v - \int v du
8494839423 \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
8572657110 1 = \int |\psi(x)|^2 dx
8572852424 \vec{E} = E(\vec{r},t)
8575746378 \int \frac{1}{2} dx = \frac{1}{2} x
8575748999 \frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x)\right) = -k^2 \left(a \sin(kx) + b \cos(kx)\right)
8576785890 1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
8577275751 0 = a \sin(0) + b\cos(0)
8582885111 \psi(x) = a \sin(kx) + b \cos(kx)
8582954722 x^2 + 2xh + h^2 = (x+h)^2
8849289982 \psi(x)^* = a \sin(\frac{n \pi}{W} x)
8889444440 1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right)\right)^2 dx
9059289981 \psi(x) = a \sin(k x)
9285928292 ax^2 + bx + c = 0
9291999979 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t}
9294858532 \hat{A}^+ = \hat{A}
9385938295 (x+(b/(2a)))^2 = -(c/a) + (b/(2a))^2
9393939991 \psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9393939992 \psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9394939493 \nabla^2 E(\vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r},t)
9429829482 \frac{d}{dx} y = -\sin(x) + i\cos(x)
9482113948 \frac{dy}{y} = i dx
9482438243 (\cos(x))^2 = \cos(2x)+(\sin(x))^2
9482923849 \exp(i x) = y
9482928242 \cos(2x) = (\cos(x))^2 - (\sin(x))^2
9482928243 \cos(2x)+(\sin(x))^2 = (\cos(x))^2
9482943948 \log(y) = i dx
9482948292 b = c
 
9482984922 \frac{d}{dx} y = (i\sin(x) + \cos(x)) i
9483928192 \cos(2x)+i\sin(2x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
9485384858 \nabla^2 E(\vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E(\vec{r})\exp(i \omega t)
9485747245 \sqrt{\frac{2}{W}} = a
9485747246 -\sqrt{\frac{2}{W}} = a
9492920340 y = \cos(x)+i \sin(x)
9494829190 k  
9495857278 \psi(x=W) = 0
9499428242 E(\vec{r},t) = E(\vec{r})\exp(i \omega t)
9499959299 a + k = b + k  
9582958293 x = \sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
9582958294 x+(b/(2a)) = \sqrt{(b/(2a))^2 - (c/a)}
9584821911 c + d = a  
9585727710 \psi(x=0) = 0
9596004948 x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
9848292229 dy = y i dx
9848294829 \frac{d}{dx} y = y i
9858028950 \frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
9889984281 2(\sin(x))^2 = 1-\cos(2x)
9919999981 \rho = 0
9958485859 \frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
9988949211 (\sin(x))^2 = \frac{1-\cos(2x)}{2}
9991999979 \vec{\nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
9999998870 \frac{\vec{p}}{\hbar} = \vec{k}
9999999870 \frac{p}{\hbar} = k
9999999950 \beta = 1/(k_{Boltzmann}T)  
9999999951 \langle x | k \rangle = \frac{\exp(i k x)}{\sqrt{2\pi}}  
9999999952 f(x) \delta(x-a) = f(a)
 
9999999953 \int_{-\infty}^{\infty} \delta(x) dx = 1
 
9999999954 c = 1/\sqrt{\epsilon_0 \mu_0}  
9999999955 \vec{E} = -\vec{\nabla} \Psi
 
9999999956 \vec{F} = \frac{\partial}{\partial t}\vec{p}  
9999999957 \vec{F} = -\vec{\nabla} V
 
9999999960 \hbar = h/(2 \pi)
9999999961 \frac{E}{\hbar} = \omega
9999999962 p = \hbar k
9999999963 \lambda = h/p
 
9999999964 \omega = c k  
9999999965 E = \omega \hbar
9999999966 \vec{L} = \vec{r}\times\vec{p}
 
9999999967 \vec{S} = \frac{1}{\mu_0} \vec{E}\times \vec{B}
 
9999999968 x = \frac{-b-\sqrt{b^2-4ac}}{2a}
9999999969 x = \frac{-b+\sqrt{b^2-4ac}}{2a}
9999999970 \eta_1 \sin\theta_1 = \eta_2 \sin\theta_2
 
9999999971 {\cal H} = \frac{p^2}{2m}+V
 
9999999972 {\cal H} |\psi\rangle = E |\psi\rangle
 
9999999973 \left( \Delta A \right)^2 = \langle A^2 \rangle - \langle A \rangle^2
 
9999999974 \langle \psi| \hat{A} |\psi\rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx
 
9999999975 \langle \psi| \hat{A} |\psi\rangle = \langle a \rangle
9999999976 \hat{p} = -i \hbar \frac{\partial }{\partial x}  
9999999977 [\hat{x},\hat{p}] = i \hbar  
9999999978 \vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}
 
9999999979 \vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}  
9999999980 \vec{\nabla} \cdot \vec{B} = 0
 
9999999981 \vec{\nabla} \cdot \vec{E} = \rho/\epsilon_0
9999999982 V = I R + Q/C + L \frac{\partial I}{\partial t}  
9999999983 C = V A/d
 
9999999984 Q = C V
 
9999999985 V = I R
 
9999999986 \left[cos⁡θsin⁡θ−sin⁡θcos⁡θ\right]\left[AxAy\right] = \left[Ax′Ay′\right]
 
9999999987 1 atmosphere = 101.325 Pascal
 
9999999988 1 atmosphere = 14.7 pounds/(inch^2)
 
9999999989 mass_{electron} = 511000 electronVolts/(q^2)
 
9999999990 Tesla = 10000*Gauss
 
9999999991 Pascal = Newton / (meter^2)
 
9999999992 Tesla = Newton / (Ampere*meter)
 
9999999993 Farad = Coulumb / Volt
 
9999999995 Volt = Joule / Coulumb
 
9999999996 Coulomb = Ampere / second
 
9999999997 Watt = Joule / second
 
9999999998 Joule = Newton*meter
 
9999999999 Newton = kilogram*meter/(second^2)
 
4961662865 x
9110536742 2 x
8483686863 \sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x)\right)
3470587782 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
8642992037 2
9894826550 2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \left(\exp(i x)+\exp(-i x)\right)
4257214632 \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)  
8699789241 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
9180861128 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
2405307372 \sin(2 x) = 2 \sin(x) \cos(x)
3268645065 x
9350663581 \pi
8332931442 \exp(i \pi) = \cos(\pi)+i \sin(\pi)
6885625907 \exp(i \pi) = -1 + i 0
3331824625 \exp(i \pi) = -1
4901237716 1
2501591100 \exp(i \pi) + 1 = 0
5136652623 E = KE + PE mechanical energy is the sum of the potential plus kinetic energies
7915321076 E, KE, PE  
3192374582 E_2, KE_2, PE_2
 
7875206161 E_2 = KE_2 + PE_2
4229092186 E, KE, PE  
1490821948 E_1, KE_1, PE_1
 
4303372136 E_1 = KE_1 + PE_1
5514556106 E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
8357234146 KE = \frac{1}{2} m v^2 kinetic energy Kinetic_energy
3658066792 KE_2, v_2
 
8082557839 KE, v  
7676652285 KE_2 = \frac{1}{2} m v_2^2
2790304597 KE_1, v_1
 
9880327189 KE, v  
4928007622 KE_1 = \frac{1}{2} m v_1^2
5733146966 KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
6554292307 t
4270680309 \frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
5781981178 x^2 - y^2 = (x+y)(x-y) difference of squares Difference_of_two_squares
5946759357 v_2, v_1
 
6675701064 x, y  
4648451961 v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
9356924046 \frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
9397152918 v = \frac{v_1 + v_2}{2} average velocity
2857430695 a = \frac{v_2 - v_1}{t}
acceleration
7735737409 \frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
4784793837 \frac{KE_2 - KE_1}{t} = m v a
5345738321 F = m a Newton's second law of motion Newton%27s_laws_of_motion#Newton's_second_law
2186083170 \frac{KE_2 - KE_1}{t} = v F
6715248283 PE = -F x potential energy Potential_energy
3852078149 PE_2, x_2
 
7388921188 PE, x  
2431507955 PE_2 = -F x_2
3412641898 PE_1, x_1
 
9937169749 PE, x  
4669290568 PE_1 = -F x_1
7734996511 PE_2 - PE_1 = -F ( x_2 - x_1 )
2016063530 t
7267155233 \frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
9337785146 v = \frac{x_2 - x_1}{t} average velocity
4872970974 \frac{PE_2 - PE_1}{t} = -F v
2081689540 t
2770069250 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
3591237106 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
1772416655 \frac{E_2 - E_1}{t} = v F - F v
1809909100 \frac{E_2 - E_1}{t} = 0
5778176146 t
3806977900 E_2 - E_1 = 0
5960438249 E_1
8558338742 E_2 = E_1
8945218208 \theta_{Brewster} + \theta_{refracted} = 90^{\circ} based on figure 34-27 on page 824 in 2001_HRW
9025853427 \theta_{Brewster}
1310571337 \theta_{refracted} = 90^{\circ} - \theta_{Brewster}
6450985774 n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 ) Law of Refraction eq 34-44 on page 819 in 2001_HRW
1779497048 \theta_{Brewster}, \theta_{refraction}  
7322344455 \theta_1, \theta_2
 
2575937347 n_1 \sin( \theta_{Brewster} ) = n_2 \sin( \theta_{refraction} )
7696214507 n_1 \sin( \theta_{Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{Brewster} )
8588429722 \sin( 90^{\circ} - x ) = \cos( x )
7375348852 \theta_{Brewster}
1512581563 x
6831637424 \sin( 90^{\circ} - \theta_{Brewster} ) = \cos( \theta_{Brewster} )
3061811650 n_1 \sin( \theta_{Brewster} ) = n_2 \cos( \theta_{Brewster} )
7857757625 n_1
9756089533 \sin( \theta_{Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{Brewster} )
5632428182 \cos( \theta_{Brewster} )
2768857871 \frac{\sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )} = \frac{n_2}{n_1}
4968680693 \tan( x ) = \frac{ \sin( x )}{\cos( x )}
7321695558 \theta_{Brewster}
9906920183 x
4501377629 \tan( \theta_{Brewster} ) = \frac{ \sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )}
3417126140 \tan( \theta_{Brewster} ) = \frac{ n_2 }{ n_1 }
5453995431 \arctan{ x }
6023986360 x
8495187962 \theta_{Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
9040079362 f
9565166889 T
5426308937 v = \frac{d}{t}
7476820482 C
1277713901 d
6946088325 v = \frac{C}{t}
6785303857 C = 2 \pi r
4816195267 C, r
 
2113447367 C_{\rm{Earth\ orbit}}, r_{\rm{Earth\ orbit}}
 
6348260313 C_{\rm{Earth\ orbit}} = 2 \pi r_{\rm{Earth\ orbit}}
3847777611 C, v, t  
6406487188 C_{\rm{Earth\ orbit}}, v_{\rm{Earth\ orbit}}, t_{\rm{Earth\ orbit}}
 
3046191961 v_{\rm{Earth\ orbit}} = \frac{C_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
3080027960 v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
7175416299 t_{\rm{Earth\ orbit}} = 1 {\rm{year}}
3219318145 \frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}
8721295221 t_{\rm{Earth\ orbit}} = 3.16 10^7 {\rm{seconds}}
4593428198 v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{3.16\ 10^7 {\rm seconds}}
3472836147 r_{\rm{Earth\ orbit}} = 1.496\ 10^8 {\rm km}
6998364753 v_{\rm{Earth\ orbit}} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
4180845508 v_{\rm{Earth\ orbit}} = 29.8 \frac{{\rm km}}{{\rm sec}}
4662369843 x' = \gamma (x - v t)
2983053062 x = \gamma (x' + v t')
3426941928 x = \gamma ( \gamma (x - v t) + v t' )
2096918413 x = \gamma ( \gamma x - \gamma v t + v t' )
7741202861 x = \gamma^2 x - \gamma^2 v t + \gamma v t'
7337056406 \gamma^2 x
4139999399 x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
3495403335 x
9031609275 x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
8014566709 \gamma^2 v t
9409776983 x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
2226340358 \gamma v
6417705759 \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = \gamma v t'
 
8730201316 \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'
first term was multiplied by \gamma/\gamma
1974334644 \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
5148266645 t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
4287102261 x^2 + y^2 + z^2 = c^2 t^2 describes a spherical wavefront
1201689765 x'^2 + y'^2 + z'^2 = c^2 t'^2
describes a spherical wavefront for an observer in a moving frame of reference
7057864873 y' = y
frame of reference is moving only along x direction
8515803375 z' = z
frame of reference is moving only along x direction
6153463771 3
 
4746576736 asdf
 
1027270623 minga
 
6101803533 ming
 
9231495607 a = b
 
8664264194 b + 2
 
9805063945 \gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2
4057686137 C \to C_{\rm{Earth\ orbit}}
2346150725 r \to r_{\rm{Earth\ orbit}}
9753878784 v \to v_{\rm{Earth\ orbit}}
8135396036 t \to t_{\rm{Earth\ orbit}}
2773628333 \theta_1 \to \theta_{Brewster}
7154592211 \theta_2 \to \theta_{\rm refracted}
3749492596 E \to E_1
1258245373 E \to E_2
4147101187 KE \to KE_1
3809726424 PE \to PE_1
6383056612 KE \to KE_2
5075406409 PE \to PE_2
5398681503 v \to v_1
9305761407 v \to v_2
4218009993 x \to x_1
4188639044 x \to x_2
8173074178 x \to v_2
1025759423 y \to v_1
1935543849 \gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2
1586866563 \left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)
3182633789 \gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
1916173354 -\gamma^2 v^2 + c^2 \gamma^2 = c^2
2076171250 -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0
5284610349 \gamma^2
2417941373 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
5787469164 1 - \gamma^2
1639827492 - c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
5669500954 v^2 \gamma^2
5763749235 -c^2 + c^2 \gamma^2 = v^2 \gamma^2
6408214498 c^2
2999795755 c^2 \gamma^2 = v^2 \gamma^2 + c^2
3412946408 v^2 \gamma^2
2542420160 c^2 \gamma^2 - v^2 \gamma^2 = c^2
7743841045 \gamma^2
7513513483 \gamma^2 (c^2 - v^2) = c^2
8571466509 c^2 - \gamma^2
7906112355 \gamma^2 = \frac{c^2}{c^2 - \gamma^2}
3060139639 1/2
 
1528310784 \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
8360117126 \gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}} not a physically valid result in this context
3366703541 a = \frac{v - v_0}{t} acceleration is the average change in speed over a duration
7083390553 t
4748157455 a t = v - v_0
6417359412 v_0
4798787814 a t + v_0 = v
3462972452 v = v_0 + a t
3411994811 v_{\rm ave} = \frac{d}{t}
6175547907 v_{\rm ave} = \frac{v + v_0}{2}
9897284307 \frac{d}{t} = \frac{v + v_0}{2}
8865085668 t
8706092970 d = \left(\frac{v + v_0}{2}\right)t
7011114072 d = \frac{(v_0 + a t) + v_0}{2} t
1265150401 d = \frac{2 v_0 + a t}{2} t
9658195023 d = v_0 t + \frac{1}{2} a t^2
5799753649 2
7215099603 v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
5144263777 v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
7939765107 v^2 = v_0^2 + 2 a d
6729698807 v_0
9759901995 v - v_0 = a t
2242144313 a
1967582749 t = \frac{v - v_0}{a}
5733721198 d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
5611024898 d = \frac{1}{2 a} (v^2 - v_0^2)
5542390646 2 a
8269198922 2 a d = v^2 - v_0^2
9070454719 v_0^2
4948763856 2 a d + v_0^2 = v^2
9645178657 a t
6457044853 v - a t = v_0
1259826355 d = (v - a t) t + \frac{1}{2} a t^2
4580545876 d = v t - a t^2 + \frac{1}{2} a t^2
6421241247 d = v t - \frac{1}{2} a t^2
4182362050 Z = |Z| \exp( i \theta ) Z \in \mathbb{C}
1928085940 Z^* = |Z| \exp( -i \theta )
9191880568 Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
3350830826 Z Z^* = |Z|^2
8396997949 I = | A + B |^2 intensity of two waves traveling opposite directions on same path
4437214608 Z
5623794884 A + B
2236639474 (A + B)(A + B)^* = |A + B|^2
1020854560 I = (A + B)(A + B)^*
6306552185 I = (A + B)(A^* + B^*)
8065128065 I = A A^* + B B^* + A B^* + B A^*
9761485403 Z
8710504862 A
4075539836 A A^* = |A|^2
6529120965 B
1511199318 Z
7107090465 B B^* = |B|^2
5125940051 I = |A|^2 + B B^* + A B^* + B A^*
1525861537 I = |A|^2 + |B|^2 + A B^* + B A^*
1742775076 Z \to B
7607271250 \theta \to \phi
4192519596 B = |B| \exp(i \phi)
2064205392 A
1894894315 Z
1357848476 A = |A| \exp(i \theta)
6555185548 A^* = |A| \exp(-i \theta)
4504256452 B^* = |B| \exp(-i \phi)
7621705408 I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
3085575328 I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
4935235303 x
2293352649 \theta - \phi
3660957533 \cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
3967985562 2
2700934933 2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
8497631728 I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
6774684564 \theta = \phi for coherent waves
2099546947 I = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )  
2719691582 |A| = |B| in a loop
3257276368 I = |A|^2 + |A|^2 + |A| |A| 2
 
8283354808 I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
8046208134 I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
1172039918 I_{\rm coherent} = 4 |A|^2
8602221482 \langle \cos(\theta - \phi) \rangle = 0 incoherent light source
6240206408 I_{\rm incoherent} = |A|^2 + |B|^2
6529793063 I_{\rm incoherent} = |A|^2 + |A|^2
3060393541 I_{\rm incoherent} = 2|A|^2
6556875579 \frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
8750500372 f
 
7543102525 g
 
8267133829 a = b
 
7968822469 c = d
 
3169580383 \vec{a} = \frac{d\vec{v}}{dt} acceleration is the change in speed over a duration
8602512487 \vec{a} = a_x \hat{x} + a_y \hat{y} decompose acceleration into two components
4158986868 a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
5349866551 \vec{v} = v_x \hat{x} + v_y \hat{y}
7729413831 a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
1819663717 a_x = \frac{d}{dt} v_x
8228733125 a_y = \frac{d}{dt} v_y
9707028061 a_x = 0
2741489181 a_y = -g
3270039798 2
8880467139 2
8750379055 0 = \frac{d}{dt} v_x
1977955751 -g = \frac{d}{dt} v_y
6672141531 dt
1702349646 -g dt = d v_y
8584698994 -g \int dt = \int d v_y
9973952056 -g t = v_y - v_{0, y}
4167526462 v_{0, y}
6572039835 -g t + v_{0, y} = v_y
7252338326 v_y = \frac{dy}{dt}
6204539227 -g t + v_{0, y} = \frac{dy}{dt}
1614343171 dt
8145337879 -g t dt + v_{0, y} dt = dy
8808860551 -g \int t dt + v_{0, y} \int dt = \int dy
2858549874 - \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
6098638221 y_0
2461349007 - \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
8717193282 dt
1166310428 0 dt = d v_x
2366691988 \int 0 dt = \int d v_x
1676472948 0 = v_x - v_{0, x}
1439089569 v_{0, x}
6134836751 v_{0, x} = v_x
8460820419 v_x = \frac{dx}{dt}
7455581657 v_{0, x} = \frac{dx}{dt}
8607458157 dt
1963253044 v_{0, x} dt = dx
3676159007 v_{0, x} \int dt = \int dx
9882526611 v_{0, x} t = x - x_0
3182907803 x_0
8486706976 v_{0, x} t + x_0 = x
1306360899 x = v_{0, x} t + x_0
7049769409 2
9341391925 \vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
6410818363 \theta
7391837535 \cos(\theta) = \frac{v_{0, x}}{v_0}
7376526845 \sin(\theta) = \frac{v_{0, y}}{v_0}
5868731041 v_0
6083821265 v_0 \cos(\theta) = v_{0, x}
5438722682 x = v_0 t \cos(\theta) + x_0
5620558729 v_0
8949329361 v_0 \sin(\theta) = v_{0, y}
1405465835 y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
9862900242 y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
6050070428 v_{0, x}
3274926090 t = \frac{x - x_0}{v_{0, x}}
7354529102 y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0
8406170337 y \to y_f
2403773761 t \to t_f
5379546684 y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
5373931751 t = t_f
9112191201 y_f = 0
8198310977 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
1650441634 y_0 = 0
define coordinate system such that initial height is at origin
1087417579 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
4829590294 t_f
2086924031 0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
6974054946 \frac{1}{2} g t_f
1191796961 \frac{1}{2} g t_f = v_0 \sin(\theta)
2510804451 2/g
4778077984 t_f = \frac{2 v_0 \sin(\theta)}{g}
3273630811 x \to x_f
6732786762 t \to t_f
3485125659 x_f = v_0 t_f \cos(\theta) + x_0
4370074654 t = t_f
2378095808 x_f = x_0 + d
4268085801 x_0 + d = v_0 t_f \cos(\theta) + x_0
8072682558 x_0
7233558441 d = v_0 t_f \cos(\theta)
2297105551 d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
7587034465 \theta
7214442790 x
2519058903 \sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
8922441655 d = \frac{v_0^2}{g} \sin(2 \theta)
5667870149 \theta
1541916015 \theta = \frac{\pi}{4}
3607070319 d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
5353282496 d = \frac{v_0^2}{g}
0000040490 a^2
0000999900 b/(2a)
0001030901 \cos(x)
0001111111 (\sin(x))^2
0001209482 2 \pi
0001304952 \hbar
0001334112 W
0001921933 2 i
0002239424 2
0002338514 \vec{p}_{2}
0002342425 m/m
0002367209 1/2
 
0002393922 x
0002424922 a
0002436656 i \hbar
0002449291 b/(2a)
0002838490 b/(2a)
0002919191 \sin(-x)
0002929944 1/2
0002940021 2 \pi
0003232242 t
0003413423 \cos(-x)
0003747849 -1
0003838111 2
0003919391 x
0003949052 -x
0003949921 \hbar
0003954314 dx
0003981813 -\sin(x)
0004089571 2x
0004264724 y
0004307451 (b/(2a))^2
0004582412 x
0004829194 2
0004831494 a
0004849392 x
0004858592 h
0004934845 x
0004948585 a
0005395034 a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
0005626421 t
0005749291 f
0005938585 \frac{-\hbar^2}{2m}
0006466214 (\sin(x))^2
0006544644 t
0006563727 x
0006644853 c/a
0006656532 e
0007471778 2(\sin(x))^2
0007563791 i
0007636749 x
0007894942 (\sin(x))^2
0008837284 T
0008842811 \cos(2x)
0009458842 \psi(x)
0009484724 \frac{n \pi}{W}x
0009485857 a^2\frac{2}{W}
0009485858 \frac{2n\pi}{W}
0009492929 v du
0009587738 \psi
0009594995 1/2
 
0009877781 y
0203024440 1 = \int_0^W a \sin(\frac{n \pi}{W} x) \psi(x)^* dx
0404050504 \lambda = \frac{v}{f}
0439492440 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\right|_0^W
0934990943 k = \frac{2 \pi}{v T}
0948572140 \int \cos(a x) dx = \frac{1}{a}\sin(a x)
1010393913 \langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*
1010393944 x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
1010923823 k W = n \pi
1020010291 0 = a \sin(k W)
1020394900 p = h/\lambda
1020394902 E = h f
1029039903 p = m v
1029039904 p^2 = m^2 v^2
1158485859 \frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
1202310110 \frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1202312210 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1203938249 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
1248277773 \cos(2x) = 1-2(\sin(x))^2
1293913110 0 = b
1293923844 \lambda = v T
1314464131 \vec{\nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
1314864131 \vec{\nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
1395858355 x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
1492811142 f = x - d  
1492842000 \nabla \vec{x} = f(y)  
1636453295 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = - \nabla^2 \vec{E}
1638282134 \vec{p}_{before} = \vec{p}_{after}
1648958381 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{\nabla}\psi(\vec{r},t)\right)
1857710291 0 = a \sin(n \pi)
1858578388 \nabla^2 E(\vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E(\vec{r})\exp(i \omega t)
1858772113 k = \frac{n \pi}{W}
1934748140 \int |\psi(x)|^2 dx = 1
2029293929 \nabla^2 E(\vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r})\exp(i \omega t)
2103023049 \sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right)
2113211456 f = 1/T
2123139121 -\exp(-i x) = -\cos(x)+i \sin(x)
2131616531 T f = 1
2258485859 {\cal H} \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
2394240499 x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
2394853829 \exp(-i x) = \cos(-x)+i \sin(-x)
2394935831 ( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
2394935835 \left(\langle\psi| \hat{A} |\psi\rangle \right)^+ = \left(\langle a \rangle\right)^+
2395958385 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{-p^2}{\hbar} \psi(\vec{r},t)
2404934990 \langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
2494533900 \langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
2569154141 \vec{\nabla} \times \frac{\partial}{\partial t}\vec{H} = \epsilon_0 \frac{\partial^2 }{\partial t^2}\vec{E}  
2648958382 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p}\psi(\vec{r},t)\right)
2848934890 \langle a \rangle^* = \langle a \rangle
2944838499 \psi(x) = a \sin(\frac{n \pi}{W} x)
3121234211 \frac{k}{2\pi} = \lambda
3121234212 p = \frac{h k}{2\pi}
3121513111 k = \frac{2 \pi}{\lambda}
3131111133 T = 1 / f
3131211131 \omega = 2 \pi f
3132131132 \omega = \frac{2\pi}{T}
3147472131 \frac{\omega}{2 \pi} = f
3285732911 (\cos(x))^2 = 1-(\sin(x))^2
3485475729 \nabla^2 E(\vec{r}) = - \frac{\omega^2}{c^2} E(\vec{r})
3585845894 \langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
3829492824 \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right) = \cos(x)
3924948349 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
3942849294 \exp(i x)-\exp(-i x) = 2 i \sin(x)
3943939590 x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
3947269979 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
3948571256 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi(\vec{r},t)
3948572230 \vec{\nabla}\psi(\vec{r},t) = \psi_0 \vec{\nabla}\exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)  
3948574224 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\vec{k}\cdot\vec{r} - \omega t \right) \right)
3948574226 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
3948574228 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3948574230 \psi(\vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
3948574233 \frac{\partial}{\partial t}\psi(\vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3948574235 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)  
3951205425 \vec{p}_{after} = \vec{p}_{1}
4147472132 E = \frac{h \omega}{2 \pi}
4298359835 E = \frac{1}{2}m v^2
4298359845 E = \frac{1}{2m}m^2 v^2
4298359851 E = \frac{p^2}{2m}
4341171256 i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{p^2}{2 m} \psi(\vec{r},t)
4348571256 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi(\vec{r},t)
4394958389 \vec{\nabla}\cdot \left(\vec{\nabla}\psi(\vec{r},t)\right) = \frac{i}{\hbar} \vec{\nabla}\cdot\left( \vec{p} \psi(\vec{r},t)\right)
4585828572 \epsilon_0 \mu_0 = \frac{1}{c^2}
4585932229 \cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
4598294821 \exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
4638429483 \exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
4742644828 \exp(i x)+\exp(-i x) = 2 \cos(x)
4827492911 \cos(2x)+(\sin(x))^2 = 1-(\sin(x))^2
4838429483 \exp(2 i x) = \cos(2 x)+i \sin(2 x)
4843995999 \frac{1}{2 i}\left(\exp(i x)-\exp(-i x)\right) = \sin(x)
4857472413 1 = \int \psi(x)\psi(x)^* dx
4857475848 \frac{1}{a^2} = \frac{W}{2}
4858429483 \exp(i x)\exp(i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))  
4923339482 i x = \log(y)
4928239482 \log(y) = i x
4928923942 a = b  
4929218492 a+b = c  
4929482992 b = 2
 
4938429482 \exp(-i x) = \cos(x)+i \sin(-x)
4938429483 \exp(i x) = \cos(x)+i \sin(x)
4938429484 \exp(-i x) = \cos(x)-i \sin(x)
4943571230 \vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
4948934890 \langle \psi| \hat{A} |\psi\rangle = \langle a \rangle^*
4949359835 \langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
4954839242 \cos(2x)+i\sin(2x) = (\cos(x)+i \sin(x))(\cos(x)+i \sin(x))  
4978429483 \exp(-i x) = \cos(-x)+i \sin(-x)  
4984892984 a+2 = c
 
4985825552 \nabla^2 E(\vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E(\vec{r})\exp(i \omega t)
5530148480 \vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
5727578862 \frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
5832984291 (\sin(x))^2 + (\cos(x))^2 = 1
5857434758 \int a dx = a x
5868688585 \frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = \frac{p^2}{2m} \psi(\vec{r},t)
5900595848 k = \frac{\omega}{v}
5928285821 x^2 + 2x(b/(2a)) + (b/(2a))^2 = (x+(b/(2a)))^2
5928292841 x^2 + (b/a)x + (b/(2a))^2 = -c/a + (b/(2a))^2
5938459282 x^2 + (b/a)x = -c/a
5958392859 x^2 + (b/a)x+(c/a) = 0
5959282914 x^2 + x(b/a) + (b/(2a))^2 = (x+(b/(2a)))^2
5982958248 x = -\sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
5982958249 x+(b/(2a)) = -\sqrt{(b/(2a))^2 - (c/a)}
5985371230 \vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi(\vec{r},t)
6742123016 \vec{p}_{electron}\cdot\vec{p}_{electron} = (\vec{p}_{1}\cdot\vec{p}_{1})+(\vec{p}_{2}\cdot\vec{p}_{2})-2(\vec{p}_{1}\cdot\vec{p}_{2})
7466829492 \vec{\nabla} \cdot \vec{E} = 0
7564894985 \int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
7572664728 \cos(2x)+2(\sin(x))^2 = 1
7575738420 \left(\sin\left(\frac{n \pi}{W}x\right)\right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
7575859295 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859300 \epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{\nabla} \times \vec{E} )_k = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859302 \epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859304 \epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
7575859306 \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859308 \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859310 \hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859312 \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7917051060 \vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
8139187332 \vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
8257621077 \vec{p}_{before} = \vec{p}_{1}
8311458118 \vec{p}_{after} = \vec{p}_{2}+\vec{p}_{electron}
8399484849 \langle x^2 -2x\langle x \rangle+\langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
8484544728 -a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
8485757728 a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
8485867742 \frac{2}{W} = a^2
8489593958 d(u v) = u dv + v du
8489593960 d(u v) - v du = u dv
8489593962 u dv = d(u v) - v du
8489593964 \int u dv = u v - \int v du
8494839423 \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
8572657110 1 = \int |\psi(x)|^2 dx
8572852424 \vec{E} = E(\vec{r},t)
8575746378 \int \frac{1}{2} dx = \frac{1}{2} x
8575748999 \frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x)\right) = -k^2 \left(a \sin(kx) + b \cos(kx)\right)
8576785890 1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
8577275751 0 = a \sin(0) + b\cos(0)
8582885111 \psi(x) = a \sin(kx) + b \cos(kx)
8582954722 x^2 + 2xh + h^2 = (x+h)^2
8849289982 \psi(x)^* = a \sin(\frac{n \pi}{W} x)
8889444440 1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right)\right)^2 dx
9059289981 \psi(x) = a \sin(k x)
9285928292 ax^2 + bx + c = 0
9291999979 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t}
9294858532 \hat{A}^+ = \hat{A}
9385938295 (x+(b/(2a)))^2 = -(c/a) + (b/(2a))^2
9393939991 \psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9393939992 \psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9394939493 \nabla^2 E(\vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r},t)
9429829482 \frac{d}{dx} y = -\sin(x) + i\cos(x)
9482113948 \frac{dy}{y} = i dx
9482438243 (\cos(x))^2 = \cos(2x)+(\sin(x))^2
9482923849 \exp(i x) = y
9482928242 \cos(2x) = (\cos(x))^2 - (\sin(x))^2
9482928243 \cos(2x)+(\sin(x))^2 = (\cos(x))^2
9482943948 \log(y) = i dx
9482948292 b = c
 
9482984922 \frac{d}{dx} y = (i\sin(x) + \cos(x)) i
9483928192 \cos(2x)+i\sin(2x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
9485384858 \nabla^2 E(\vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E(\vec{r})\exp(i \omega t)
9485747245 \sqrt{\frac{2}{W}} = a
9485747246 -\sqrt{\frac{2}{W}} = a
9492920340 y = \cos(x)+i \sin(x)
9494829190 k  
9495857278 \psi(x=W) = 0
9499428242 E(\vec{r},t) = E(\vec{r})\exp(i \omega t)
9499959299 a + k = b + k  
9582958293 x = \sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
9582958294 x+(b/(2a)) = \sqrt{(b/(2a))^2 - (c/a)}
9584821911 c + d = a  
9585727710 \psi(x=0) = 0
9596004948 x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
9848292229 dy = y i dx
9848294829 \frac{d}{dx} y = y i
9858028950 \frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
9889984281 2(\sin(x))^2 = 1-\cos(2x)
9919999981 \rho = 0
9958485859 \frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
9988949211 (\sin(x))^2 = \frac{1-\cos(2x)}{2}
9991999979 \vec{\nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
9999998870 \frac{\vec{p}}{\hbar} = \vec{k}
9999999870 \frac{p}{\hbar} = k
9999999950 \beta = 1/(k_{Boltzmann}T)  
9999999951 \langle x | k \rangle = \frac{\exp(i k x)}{\sqrt{2\pi}}  
9999999952 f(x) \delta(x-a) = f(a)
 
9999999953 \int_{-\infty}^{\infty} \delta(x) dx = 1
 
9999999954 c = 1/\sqrt{\epsilon_0 \mu_0}  
9999999955 \vec{E} = -\vec{\nabla} \Psi
 
9999999956 \vec{F} = \frac{\partial}{\partial t}\vec{p}  
9999999957 \vec{F} = -\vec{\nabla} V
 
9999999960 \hbar = h/(2 \pi)
9999999961 \frac{E}{\hbar} = \omega
9999999962 p = \hbar k
9999999963 \lambda = h/p
 
9999999964 \omega = c k  
9999999965 E = \omega \hbar
9999999966 \vec{L} = \vec{r}\times\vec{p}
 
9999999967 \vec{S} = \frac{1}{\mu_0} \vec{E}\times \vec{B}
 
9999999968 x = \frac{-b-\sqrt{b^2-4ac}}{2a}
9999999969 x = \frac{-b+\sqrt{b^2-4ac}}{2a}
9999999970 \eta_1 \sin\theta_1 = \eta_2 \sin\theta_2
 
9999999971 {\cal H} = \frac{p^2}{2m}+V
 
9999999972 {\cal H} |\psi\rangle = E |\psi\rangle
 
9999999973 \left( \Delta A \right)^2 = \langle A^2 \rangle - \langle A \rangle^2
 
9999999974 \langle \psi| \hat{A} |\psi\rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx
 
9999999975 \langle \psi| \hat{A} |\psi\rangle = \langle a \rangle
9999999976 \hat{p} = -i \hbar \frac{\partial }{\partial x}  
9999999977 [\hat{x},\hat{p}] = i \hbar  
9999999978 \vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}
 
9999999979 \vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}  
9999999980 \vec{\nabla} \cdot \vec{B} = 0
 
9999999981 \vec{\nabla} \cdot \vec{E} = \rho/\epsilon_0
9999999982 V = I R + Q/C + L \frac{\partial I}{\partial t}  
9999999983 C = V A/d
 
9999999984 Q = C V
 
9999999985 V = I R
 
9999999986 \left[cos⁡θsin⁡θ−sin⁡θcos⁡θ\right]\left[AxAy\right] = \left[Ax′Ay′\right]
 
9999999987 1 atmosphere = 101.325 Pascal
 
9999999988 1 atmosphere = 14.7 pounds/(inch^2)
 
9999999989 mass_{electron} = 511000 electronVolts/(q^2)
 
9999999990 Tesla = 10000*Gauss
 
9999999991 Pascal = Newton / (meter^2)
 
9999999992 Tesla = Newton / (Ampere*meter)
 
9999999993 Farad = Coulumb / Volt
 
9999999995 Volt = Joule / Coulumb
 
9999999996 Coulomb = Ampere / second
 
9999999997 Watt = Joule / second
 
9999999998 Joule = Newton*meter
 
9999999999 Newton = kilogram*meter/(second^2)
 
4961662865 x
9110536742 2 x
8483686863 \sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x)\right)
3470587782 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
8642992037 2
9894826550 2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \left(\exp(i x)+\exp(-i x)\right)
4257214632 \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)  
8699789241 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
9180861128 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
2405307372 \sin(2 x) = 2 \sin(x) \cos(x)
3268645065 x
9350663581 \pi
8332931442 \exp(i \pi) = \cos(\pi)+i \sin(\pi)
6885625907 \exp(i \pi) = -1 + i 0
3331824625 \exp(i \pi) = -1
4901237716 1
2501591100 \exp(i \pi) + 1 = 0
5136652623 E = KE + PE mechanical energy is the sum of the potential plus kinetic energies
7915321076 E, KE, PE  
3192374582 E_2, KE_2, PE_2
 
7875206161 E_2 = KE_2 + PE_2
4229092186 E, KE, PE  
1490821948 E_1, KE_1, PE_1
 
4303372136 E_1 = KE_1 + PE_1
5514556106 E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
8357234146 KE = \frac{1}{2} m v^2 kinetic energy Kinetic_energy
3658066792 KE_2, v_2
 
8082557839 KE, v  
7676652285 KE_2 = \frac{1}{2} m v_2^2
2790304597 KE_1, v_1
 
9880327189 KE, v  
4928007622 KE_1 = \frac{1}{2} m v_1^2
5733146966 KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
6554292307 t
4270680309 \frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
5781981178 x^2 - y^2 = (x+y)(x-y) difference of squares Difference_of_two_squares
5946759357 v_2, v_1
 
6675701064 x, y  
4648451961 v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
9356924046 \frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
9397152918 v = \frac{v_1 + v_2}{2} average velocity
2857430695 a = \frac{v_2 - v_1}{t}
acceleration
7735737409 \frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
4784793837 \frac{KE_2 - KE_1}{t} = m v a
5345738321 F = m a Newton's second law of motion Newton%27s_laws_of_motion#Newton's_second_law
2186083170 \frac{KE_2 - KE_1}{t} = v F
6715248283 PE = -F x potential energy Potential_energy
3852078149 PE_2, x_2
 
7388921188 PE, x  
2431507955 PE_2 = -F x_2
3412641898 PE_1, x_1
 
9937169749 PE, x  
4669290568 PE_1 = -F x_1
7734996511 PE_2 - PE_1 = -F ( x_2 - x_1 )
2016063530 t
7267155233 \frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
9337785146 v = \frac{x_2 - x_1}{t} average velocity
4872970974 \frac{PE_2 - PE_1}{t} = -F v
2081689540 t
2770069250 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
3591237106 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
1772416655 \frac{E_2 - E_1}{t} = v F - F v
1809909100 \frac{E_2 - E_1}{t} = 0
5778176146 t
3806977900 E_2 - E_1 = 0
5960438249 E_1
8558338742 E_2 = E_1
8945218208 \theta_{Brewster} + \theta_{refracted} = 90^{\circ} based on figure 34-27 on page 824 in 2001_HRW
9025853427 \theta_{Brewster}
1310571337 \theta_{refracted} = 90^{\circ} - \theta_{Brewster}
6450985774 n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 ) Law of Refraction eq 34-44 on page 819 in 2001_HRW
1779497048 \theta_{Brewster}, \theta_{refraction}  
7322344455 \theta_1, \theta_2
 
2575937347 n_1 \sin( \theta_{Brewster} ) = n_2 \sin( \theta_{refraction} )
7696214507 n_1 \sin( \theta_{Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{Brewster} )
8588429722 \sin( 90^{\circ} - x ) = \cos( x )
7375348852 \theta_{Brewster}
1512581563 x
6831637424 \sin( 90^{\circ} - \theta_{Brewster} ) = \cos( \theta_{Brewster} )
3061811650 n_1 \sin( \theta_{Brewster} ) = n_2 \cos( \theta_{Brewster} )
7857757625 n_1
9756089533 \sin( \theta_{Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{Brewster} )
5632428182 \cos( \theta_{Brewster} )
2768857871 \frac{\sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )} = \frac{n_2}{n_1}
4968680693 \tan( x ) = \frac{ \sin( x )}{\cos( x )}
7321695558 \theta_{Brewster}
9906920183 x
4501377629 \tan( \theta_{Brewster} ) = \frac{ \sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )}
3417126140 \tan( \theta_{Brewster} ) = \frac{ n_2 }{ n_1 }
5453995431 \arctan{ x }
6023986360 x
8495187962 \theta_{Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
9040079362 f
9565166889 T
5426308937 v = \frac{d}{t}
7476820482 C
1277713901 d
6946088325 v = \frac{C}{t}
6785303857 C = 2 \pi r
4816195267 C, r
 
2113447367 C_{\rm{Earth\ orbit}}, r_{\rm{Earth\ orbit}}
 
6348260313 C_{\rm{Earth\ orbit}} = 2 \pi r_{\rm{Earth\ orbit}}
3847777611 C, v, t  
6406487188 C_{\rm{Earth\ orbit}}, v_{\rm{Earth\ orbit}}, t_{\rm{Earth\ orbit}}
 
3046191961 v_{\rm{Earth\ orbit}} = \frac{C_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
3080027960 v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
7175416299 t_{\rm{Earth\ orbit}} = 1 {\rm{year}}
3219318145 \frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}
8721295221 t_{\rm{Earth\ orbit}} = 3.16 10^7 {\rm{seconds}}
4593428198 v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{3.16\ 10^7 {\rm seconds}}
3472836147 r_{\rm{Earth\ orbit}} = 1.496\ 10^8 {\rm km}
6998364753 v_{\rm{Earth\ orbit}} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
4180845508 v_{\rm{Earth\ orbit}} = 29.8 \frac{{\rm km}}{{\rm sec}}
4662369843 x' = \gamma (x - v t)
2983053062 x = \gamma (x' + v t')
3426941928 x = \gamma ( \gamma (x - v t) + v t' )
2096918413 x = \gamma ( \gamma x - \gamma v t + v t' )
7741202861 x = \gamma^2 x - \gamma^2 v t + \gamma v t'
7337056406 \gamma^2 x
4139999399 x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
3495403335 x
9031609275 x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
8014566709 \gamma^2 v t
9409776983 x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
2226340358 \gamma v
6417705759 \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = \gamma v t'
 
8730201316 \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'
first term was multiplied by \gamma/\gamma
1974334644 \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
5148266645 t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
4287102261 x^2 + y^2 + z^2 = c^2 t^2 describes a spherical wavefront
1201689765 x'^2 + y'^2 + z'^2 = c^2 t'^2
describes a spherical wavefront for an observer in a moving frame of reference
7057864873 y' = y
frame of reference is moving only along x direction
8515803375 z' = z
frame of reference is moving only along x direction
6153463771 3
 
4746576736 asdf
 
1027270623 minga
 
6101803533 ming
 
9231495607 a = b
 
8664264194 b + 2
 
9805063945 \gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2
4057686137 C \to C_{\rm{Earth\ orbit}}
2346150725 r \to r_{\rm{Earth\ orbit}}
9753878784 v \to v_{\rm{Earth\ orbit}}
8135396036 t \to t_{\rm{Earth\ orbit}}
2773628333 \theta_1 \to \theta_{Brewster}
7154592211 \theta_2 \to \theta_{\rm refracted}
3749492596 E \to E_1
1258245373 E \to E_2
4147101187 KE \to KE_1
3809726424 PE \to PE_1
6383056612 KE \to KE_2
5075406409 PE \to PE_2
5398681503 v \to v_1
9305761407 v \to v_2
4218009993 x \to x_1
4188639044 x \to x_2
8173074178 x \to v_2
1025759423 y \to v_1
1935543849 \gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2
1586866563 \left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)
3182633789 \gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
1916173354 -\gamma^2 v^2 + c^2 \gamma^2 = c^2
2076171250 -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0
5284610349 \gamma^2
2417941373 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
5787469164 1 - \gamma^2
1639827492 - c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
5669500954 v^2 \gamma^2
5763749235 -c^2 + c^2 \gamma^2 = v^2 \gamma^2
6408214498 c^2
2999795755 c^2 \gamma^2 = v^2 \gamma^2 + c^2
3412946408 v^2 \gamma^2
2542420160 c^2 \gamma^2 - v^2 \gamma^2 = c^2
7743841045 \gamma^2
7513513483 \gamma^2 (c^2 - v^2) = c^2
8571466509 c^2 - \gamma^2
7906112355 \gamma^2 = \frac{c^2}{c^2 - \gamma^2}
3060139639 1/2
 
1528310784 \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
8360117126 \gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}} not a physically valid result in this context
3366703541 a = \frac{v - v_0}{t} acceleration is the average change in speed over a duration
7083390553 t
4748157455 a t = v - v_0
6417359412 v_0
4798787814 a t + v_0 = v
3462972452 v = v_0 + a t
3411994811 v_{\rm ave} = \frac{d}{t}
6175547907 v_{\rm ave} = \frac{v + v_0}{2}
9897284307 \frac{d}{t} = \frac{v + v_0}{2}
8865085668 t
8706092970 d = \left(\frac{v + v_0}{2}\right)t
7011114072 d = \frac{(v_0 + a t) + v_0}{2} t
1265150401 d = \frac{2 v_0 + a t}{2} t
9658195023 d = v_0 t + \frac{1}{2} a t^2
5799753649 2
7215099603 v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
5144263777 v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
7939765107 v^2 = v_0^2 + 2 a d
6729698807 v_0
9759901995 v - v_0 = a t
2242144313 a
1967582749 t = \frac{v - v_0}{a}
5733721198 d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
5611024898 d = \frac{1}{2 a} (v^2 - v_0^2)
5542390646 2 a
8269198922 2 a d = v^2 - v_0^2
9070454719 v_0^2
4948763856 2 a d + v_0^2 = v^2
9645178657 a t
6457044853 v - a t = v_0
1259826355 d = (v - a t) t + \frac{1}{2} a t^2
4580545876 d = v t - a t^2 + \frac{1}{2} a t^2
6421241247 d = v t - \frac{1}{2} a t^2
4182362050 Z = |Z| \exp( i \theta ) Z \in \mathbb{C}
1928085940 Z^* = |Z| \exp( -i \theta )
9191880568 Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
3350830826 Z Z^* = |Z|^2
8396997949 I = | A + B |^2 intensity of two waves traveling opposite directions on same path
4437214608 Z
5623794884 A + B
2236639474 (A + B)(A + B)^* = |A + B|^2
1020854560 I = (A + B)(A + B)^*
6306552185 I = (A + B)(A^* + B^*)
8065128065 I = A A^* + B B^* + A B^* + B A^*
9761485403 Z
8710504862 A
4075539836 A A^* = |A|^2
6529120965 B
1511199318 Z
7107090465 B B^* = |B|^2
5125940051 I = |A|^2 + B B^* + A B^* + B A^*
1525861537 I = |A|^2 + |B|^2 + A B^* + B A^*
1742775076 Z \to B
7607271250 \theta \to \phi
4192519596 B = |B| \exp(i \phi)
2064205392 A
1894894315 Z
1357848476 A = |A| \exp(i \theta)
6555185548 A^* = |A| \exp(-i \theta)
4504256452 B^* = |B| \exp(-i \phi)
7621705408 I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
3085575328 I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
4935235303 x
2293352649 \theta - \phi
3660957533 \cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
3967985562 2
2700934933 2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
8497631728 I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
6774684564 \theta = \phi for coherent waves
2099546947 I = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )  
2719691582 |A| = |B| in a loop
3257276368 I = |A|^2 + |A|^2 + |A| |A| 2
 
8283354808 I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
8046208134 I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
1172039918 I_{\rm coherent} = 4 |A|^2
8602221482 \langle \cos(\theta - \phi) \rangle = 0 incoherent light source
6240206408 I_{\rm incoherent} = |A|^2 + |B|^2
6529793063 I_{\rm incoherent} = |A|^2 + |A|^2
3060393541 I_{\rm incoherent} = 2|A|^2
6556875579 \frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
8750500372 f
 
7543102525 g
 
8267133829 a = b
 
7968822469 c = d
 
3169580383 \vec{a} = \frac{d\vec{v}}{dt} acceleration is the change in speed over a duration
8602512487 \vec{a} = a_x \hat{x} + a_y \hat{y} decompose acceleration into two components
4158986868 a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
5349866551 \vec{v} = v_x \hat{x} + v_y \hat{y}
7729413831 a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
1819663717 a_x = \frac{d}{dt} v_x
8228733125 a_y = \frac{d}{dt} v_y
9707028061 a_x = 0
2741489181 a_y = -g
3270039798 2
8880467139 2
8750379055 0 = \frac{d}{dt} v_x
1977955751 -g = \frac{d}{dt} v_y
6672141531 dt
1702349646 -g dt = d v_y
8584698994 -g \int dt = \int d v_y
9973952056 -g t = v_y - v_{0, y}
4167526462 v_{0, y}
6572039835 -g t + v_{0, y} = v_y
7252338326 v_y = \frac{dy}{dt}
6204539227 -g t + v_{0, y} = \frac{dy}{dt}
1614343171 dt
8145337879 -g t dt + v_{0, y} dt = dy
8808860551 -g \int t dt + v_{0, y} \int dt = \int dy
2858549874 - \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
6098638221 y_0
2461349007 - \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
8717193282 dt
1166310428 0 dt = d v_x
2366691988 \int 0 dt = \int d v_x
1676472948 0 = v_x - v_{0, x}
1439089569 v_{0, x}
6134836751 v_{0, x} = v_x
8460820419 v_x = \frac{dx}{dt}
7455581657 v_{0, x} = \frac{dx}{dt}
8607458157 dt
1963253044 v_{0, x} dt = dx
3676159007 v_{0, x} \int dt = \int dx
9882526611 v_{0, x} t = x - x_0
3182907803 x_0
8486706976 v_{0, x} t + x_0 = x
1306360899 x = v_{0, x} t + x_0
7049769409 2
9341391925 \vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
6410818363 \theta
7391837535 \cos(\theta) = \frac{v_{0, x}}{v_0}
7376526845 \sin(\theta) = \frac{v_{0, y}}{v_0}
5868731041 v_0
6083821265 v_0 \cos(\theta) = v_{0, x}
5438722682 x = v_0 t \cos(\theta) + x_0
5620558729 v_0
8949329361 v_0 \sin(\theta) = v_{0, y}
1405465835 y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
9862900242 y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
6050070428 v_{0, x}
3274926090 t = \frac{x - x_0}{v_{0, x}}
7354529102 y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0
8406170337 y \to y_f
2403773761 t \to t_f
5379546684 y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
5373931751 t = t_f
9112191201 y_f = 0
8198310977 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
1650441634 y_0 = 0
define coordinate system such that initial height is at origin
1087417579 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
4829590294 t_f
2086924031 0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
6974054946 \frac{1}{2} g t_f
1191796961 \frac{1}{2} g t_f = v_0 \sin(\theta)
2510804451 2/g
4778077984 t_f = \frac{2 v_0 \sin(\theta)}{g}
3273630811 x \to x_f
6732786762 t \to t_f
3485125659 x_f = v_0 t_f \cos(\theta) + x_0
4370074654 t = t_f
2378095808 x_f = x_0 + d
4268085801 x_0 + d = v_0 t_f \cos(\theta) + x_0
8072682558 x_0
7233558441 d = v_0 t_f \cos(\theta)
2297105551 d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
7587034465 \theta
7214442790 x
2519058903 \sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
8922441655 d = \frac{v_0^2}{g} \sin(2 \theta)
5667870149 \theta
1541916015 \theta = \frac{\pi}{4}
3607070319 d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
5353282496 d = \frac{v_0^2}{g}
0000040490 a^2
0000999900 b/(2a)
0001030901 \cos(x)
0001111111 (\sin(x))^2
0001209482 2 \pi
0001304952 \hbar
0001334112 W
0001921933 2 i
0002239424 2
0002338514 \vec{p}_{2}
0002342425 m/m
0002367209 1/2
 
0002393922 x
0002424922 a
0002436656 i \hbar
0002449291 b/(2a)
0002838490 b/(2a)
0002919191 \sin(-x)
0002929944 1/2
0002940021 2 \pi
0003232242 t
0003413423 \cos(-x)
0003747849 -1
0003838111 2
0003919391 x
0003949052 -x
0003949921 \hbar
0003954314 dx
0003981813 -\sin(x)
0004089571 2x
0004264724 y
0004307451 (b/(2a))^2
0004582412 x
0004829194 2
0004831494 a
0004849392 x
0004858592 h
0004934845 x
0004948585 a
0005395034 a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
0005626421 t
0005749291 f
0005938585 \frac{-\hbar^2}{2m}
0006466214 (\sin(x))^2
0006544644 t
0006563727 x
0006644853 c/a
0006656532 e
0007471778 2(\sin(x))^2
0007563791 i
0007636749 x
0007894942 (\sin(x))^2
0008837284 T
0008842811 \cos(2x)
0009458842 \psi(x)
0009484724 \frac{n \pi}{W}x
0009485857 a^2\frac{2}{W}
0009485858 \frac{2n\pi}{W}
0009492929 v du
0009587738 \psi
0009594995 1/2
 
0009877781 y
0203024440 1 = \int_0^W a \sin(\frac{n \pi}{W} x) \psi(x)^* dx
0404050504 \lambda = \frac{v}{f}
0439492440 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\right|_0^W
0934990943 k = \frac{2 \pi}{v T}
0948572140 \int \cos(a x) dx = \frac{1}{a}\sin(a x)
1010393913 \langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*
1010393944 x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
1010923823 k W = n \pi
1020010291 0 = a \sin(k W)
1020394900 p = h/\lambda
1020394902 E = h f
1029039903 p = m v
1029039904 p^2 = m^2 v^2
1158485859 \frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
1202310110 \frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1202312210 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1203938249 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
1248277773 \cos(2x) = 1-2(\sin(x))^2
1293913110 0 = b
1293923844 \lambda = v T
1314464131 \vec{\nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
1314864131 \vec{\nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
1395858355 x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
1492811142 f = x - d  
1492842000 \nabla \vec{x} = f(y)  
1636453295 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = - \nabla^2 \vec{E}
1638282134 \vec{p}_{before} = \vec{p}_{after}
1648958381 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{\nabla}\psi(\vec{r},t)\right)
1857710291 0 = a \sin(n \pi)
1858578388 \nabla^2 E(\vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E(\vec{r})\exp(i \omega t)
1858772113 k = \frac{n \pi}{W}
1934748140 \int |\psi(x)|^2 dx = 1
2029293929 \nabla^2 E(\vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r})\exp(i \omega t)
2103023049 \sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right)
2113211456 f = 1/T
2123139121 -\exp(-i x) = -\cos(x)+i \sin(x)
2131616531 T f = 1
2258485859 {\cal H} \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
2394240499 x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
2394853829 \exp(-i x) = \cos(-x)+i \sin(-x)
2394935831 ( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
2394935835 \left(\langle\psi| \hat{A} |\psi\rangle \right)^+ = \left(\langle a \rangle\right)^+
2395958385 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{-p^2}{\hbar} \psi(\vec{r},t)
2404934990 \langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
2494533900 \langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
2569154141 \vec{\nabla} \times \frac{\partial}{\partial t}\vec{H} = \epsilon_0 \frac{\partial^2 }{\partial t^2}\vec{E}  
2648958382 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p}\psi(\vec{r},t)\right)
2848934890 \langle a \rangle^* = \langle a \rangle
2944838499 \psi(x) = a \sin(\frac{n \pi}{W} x)
3121234211 \frac{k}{2\pi} = \lambda
3121234212 p = \frac{h k}{2\pi}
3121513111 k = \frac{2 \pi}{\lambda}
3131111133 T = 1 / f
3131211131 \omega = 2 \pi f
3132131132 \omega = \frac{2\pi}{T}
3147472131 \frac{\omega}{2 \pi} = f
3285732911 (\cos(x))^2 = 1-(\sin(x))^2
3485475729 \nabla^2 E(\vec{r}) = - \frac{\omega^2}{c^2} E(\vec{r})
3585845894 \langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
3829492824 \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right) = \cos(x)
3924948349 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
3942849294 \exp(i x)-\exp(-i x) = 2 i \sin(x)
3943939590 x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
3947269979 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
3948571256 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi(\vec{r},t)
3948572230 \vec{\nabla}\psi(\vec{r},t) = \psi_0 \vec{\nabla}\exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)  
3948574224 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\vec{k}\cdot\vec{r} - \omega t \right) \right)
3948574226 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
3948574228 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3948574230 \psi(\vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
3948574233 \frac{\partial}{\partial t}\psi(\vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3948574235 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)  
3951205425 \vec{p}_{after} = \vec{p}_{1}
4147472132 E = \frac{h \omega}{2 \pi}
4298359835 E = \frac{1}{2}m v^2
4298359845 E = \frac{1}{2m}m^2 v^2
4298359851 E = \frac{p^2}{2m}
4341171256 i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{p^2}{2 m} \psi(\vec{r},t)
4348571256 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi(\vec{r},t)
4394958389 \vec{\nabla}\cdot \left(\vec{\nabla}\psi(\vec{r},t)\right) = \frac{i}{\hbar} \vec{\nabla}\cdot\left( \vec{p} \psi(\vec{r},t)\right)
4585828572 \epsilon_0 \mu_0 = \frac{1}{c^2}
4585932229 \cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
4598294821 \exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
4638429483 \exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
4742644828 \exp(i x)+\exp(-i x) = 2 \cos(x)
4827492911 \cos(2x)+(\sin(x))^2 = 1-(\sin(x))^2
4838429483 \exp(2 i x) = \cos(2 x)+i \sin(2 x)
4843995999 \frac{1}{2 i}\left(\exp(i x)-\exp(-i x)\right) = \sin(x)
4857472413 1 = \int \psi(x)\psi(x)^* dx
4857475848 \frac{1}{a^2} = \frac{W}{2}
4858429483 \exp(i x)\exp(i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))  
4923339482 i x = \log(y)
4928239482 \log(y) = i x
4928923942 a = b  
4929218492 a+b = c  
4929482992 b = 2
 
4938429482 \exp(-i x) = \cos(x)+i \sin(-x)
4938429483 \exp(i x) = \cos(x)+i \sin(x)
4938429484 \exp(-i x) = \cos(x)-i \sin(x)
4943571230 \vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
4948934890 \langle \psi| \hat{A} |\psi\rangle = \langle a \rangle^*
4949359835 \langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
4954839242 \cos(2x)+i\sin(2x) = (\cos(x)+i \sin(x))(\cos(x)+i \sin(x))  
4978429483 \exp(-i x) = \cos(-x)+i \sin(-x)  
4984892984 a+2 = c
 
4985825552 \nabla^2 E(\vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E(\vec{r})\exp(i \omega t)
5530148480 \vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
5727578862 \frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
5832984291 (\sin(x))^2 + (\cos(x))^2 = 1
5857434758 \int a dx = a x
5868688585 \frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = \frac{p^2}{2m} \psi(\vec{r},t)
5900595848 k = \frac{\omega}{v}
5928285821 x^2 + 2x(b/(2a)) + (b/(2a))^2 = (x+(b/(2a)))^2
5928292841 x^2 + (b/a)x + (b/(2a))^2 = -c/a + (b/(2a))^2
5938459282 x^2 + (b/a)x = -c/a
5958392859 x^2 + (b/a)x+(c/a) = 0
5959282914 x^2 + x(b/a) + (b/(2a))^2 = (x+(b/(2a)))^2
5982958248 x = -\sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
5982958249 x+(b/(2a)) = -\sqrt{(b/(2a))^2 - (c/a)}
5985371230 \vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi(\vec{r},t)
6742123016 \vec{p}_{electron}\cdot\vec{p}_{electron} = (\vec{p}_{1}\cdot\vec{p}_{1})+(\vec{p}_{2}\cdot\vec{p}_{2})-2(\vec{p}_{1}\cdot\vec{p}_{2})
7466829492 \vec{\nabla} \cdot \vec{E} = 0
7564894985 \int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
7572664728 \cos(2x)+2(\sin(x))^2 = 1
7575738420 \left(\sin\left(\frac{n \pi}{W}x\right)\right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
7575859295 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859300 \epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{\nabla} \times \vec{E} )_k = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859302 \epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859304 \epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
7575859306 \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859308 \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859310 \hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859312 \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7917051060 \vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
8139187332 \vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
8257621077 \vec{p}_{before} = \vec{p}_{1}
8311458118 \vec{p}_{after} = \vec{p}_{2}+\vec{p}_{electron}
8399484849 \langle x^2 -2x\langle x \rangle+\langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
8484544728 -a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
8485757728 a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
8485867742 \frac{2}{W} = a^2
8489593958 d(u v) = u dv + v du
8489593960 d(u v) - v du = u dv
8489593962 u dv = d(u v) - v du
8489593964 \int u dv = u v - \int v du
8494839423 \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
8572657110 1 = \int |\psi(x)|^2 dx
8572852424 \vec{E} = E(\vec{r},t)
8575746378 \int \frac{1}{2} dx = \frac{1}{2} x
8575748999 \frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x)\right) = -k^2 \left(a \sin(kx) + b \cos(kx)\right)
8576785890 1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
8577275751 0 = a \sin(0) + b\cos(0)
8582885111 \psi(x) = a \sin(kx) + b \cos(kx)
8582954722 x^2 + 2xh + h^2 = (x+h)^2
8849289982 \psi(x)^* = a \sin(\frac{n \pi}{W} x)
8889444440 1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right)\right)^2 dx
9059289981 \psi(x) = a \sin(k x)
9285928292 ax^2 + bx + c = 0
9291999979 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t}
9294858532 \hat{A}^+ = \hat{A}
9385938295 (x+(b/(2a)))^2 = -(c/a) + (b/(2a))^2
9393939991 \psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9393939992 \psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9394939493 \nabla^2 E(\vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r},t)
9429829482 \frac{d}{dx} y = -\sin(x) + i\cos(x)
9482113948 \frac{dy}{y} = i dx
9482438243 (\cos(x))^2 = \cos(2x)+(\sin(x))^2
9482923849 \exp(i x) = y
9482928242 \cos(2x) = (\cos(x))^2 - (\sin(x))^2
9482928243 \cos(2x)+(\sin(x))^2 = (\cos(x))^2
9482943948 \log(y) = i dx
9482948292 b = c
 
9482984922 \frac{d}{dx} y = (i\sin(x) + \cos(x)) i
9483928192 \cos(2x)+i\sin(2x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
9485384858 \nabla^2 E(\vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E(\vec{r})\exp(i \omega t)
9485747245 \sqrt{\frac{2}{W}} = a
9485747246 -\sqrt{\frac{2}{W}} = a
9492920340 y = \cos(x)+i \sin(x)
9494829190 k  
9495857278 \psi(x=W) = 0
9499428242 E(\vec{r},t) = E(\vec{r})\exp(i \omega t)
9499959299 a + k = b + k  
9582958293 x = \sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
9582958294 x+(b/(2a)) = \sqrt{(b/(2a))^2 - (c/a)}
9584821911 c + d = a  
9585727710 \psi(x=0) = 0
9596004948 x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
9848292229 dy = y i dx
9848294829 \frac{d}{dx} y = y i
9858028950 \frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
9889984281 2(\sin(x))^2 = 1-\cos(2x)
9919999981 \rho = 0
9958485859 \frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
9988949211 (\sin(x))^2 = \frac{1-\cos(2x)}{2}
9991999979 \vec{\nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
9999998870 \frac{\vec{p}}{\hbar} = \vec{k}
9999999870 \frac{p}{\hbar} = k
9999999950 \beta = 1/(k_{Boltzmann}T)  
9999999951 \langle x | k \rangle = \frac{\exp(i k x)}{\sqrt{2\pi}}  
9999999952 f(x) \delta(x-a) = f(a)
 
9999999953 \int_{-\infty}^{\infty} \delta(x) dx = 1
 
9999999954 c = 1/\sqrt{\epsilon_0 \mu_0}  
9999999955 \vec{E} = -\vec{\nabla} \Psi
 
9999999956 \vec{F} = \frac{\partial}{\partial t}\vec{p}  
9999999957 \vec{F} = -\vec{\nabla} V
 
9999999960 \hbar = h/(2 \pi)
9999999961 \frac{E}{\hbar} = \omega
9999999962 p = \hbar k
9999999963 \lambda = h/p
 
9999999964 \omega = c k  
9999999965 E = \omega \hbar
9999999966 \vec{L} = \vec{r}\times\vec{p}
 
9999999967 \vec{S} = \frac{1}{\mu_0} \vec{E}\times \vec{B}
 
9999999968 x = \frac{-b-\sqrt{b^2-4ac}}{2a}
9999999969 x = \frac{-b+\sqrt{b^2-4ac}}{2a}
9999999970 \eta_1 \sin\theta_1 = \eta_2 \sin\theta_2
 
9999999971 {\cal H} = \frac{p^2}{2m}+V
 
9999999972 {\cal H} |\psi\rangle = E |\psi\rangle
 
9999999973 \left( \Delta A \right)^2 = \langle A^2 \rangle - \langle A \rangle^2
 
9999999974 \langle \psi| \hat{A} |\psi\rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx
 
9999999975 \langle \psi| \hat{A} |\psi\rangle = \langle a \rangle
9999999976 \hat{p} = -i \hbar \frac{\partial }{\partial x}  
9999999977 [\hat{x},\hat{p}] = i \hbar  
9999999978 \vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}
 
9999999979 \vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}  
9999999980 \vec{\nabla} \cdot \vec{B} = 0
 
9999999981 \vec{\nabla} \cdot \vec{E} = \rho/\epsilon_0
9999999982 V = I R + Q/C + L \frac{\partial I}{\partial t}  
9999999983 C = V A/d
 
9999999984 Q = C V
 
9999999985 V = I R
 
9999999986 \left[cos⁡θsin⁡θ−sin⁡θcos⁡θ\right]\left[AxAy\right] = \left[Ax′Ay′\right]
 
9999999987 1 atmosphere = 101.325 Pascal
 
9999999988 1 atmosphere = 14.7 pounds/(inch^2)
 
9999999989 mass_{electron} = 511000 electronVolts/(q^2)
 
9999999990 Tesla = 10000*Gauss
 
9999999991 Pascal = Newton / (meter^2)
 
9999999992 Tesla = Newton / (Ampere*meter)
 
9999999993 Farad = Coulumb / Volt
 
9999999995 Volt = Joule / Coulumb
 
9999999996 Coulomb = Ampere / second
 
9999999997 Watt = Joule / second
 
9999999998 Joule = Newton*meter
 
9999999999 Newton = kilogram*meter/(second^2)
 
4961662865 x
9110536742 2 x
8483686863 \sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x)\right)
3470587782 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
8642992037 2
9894826550 2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \left(\exp(i x)+\exp(-i x)\right)
4257214632 \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)  
8699789241 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
9180861128 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
2405307372 \sin(2 x) = 2 \sin(x) \cos(x)
3268645065 x
9350663581 \pi
8332931442 \exp(i \pi) = \cos(\pi)+i \sin(\pi)
6885625907 \exp(i \pi) = -1 + i 0
3331824625 \exp(i \pi) = -1
4901237716 1
2501591100 \exp(i \pi) + 1 = 0
5136652623 E = KE + PE mechanical energy is the sum of the potential plus kinetic energies
7915321076 E, KE, PE  
3192374582 E_2, KE_2, PE_2
 
7875206161 E_2 = KE_2 + PE_2
4229092186 E, KE, PE  
1490821948 E_1, KE_1, PE_1
 
4303372136 E_1 = KE_1 + PE_1
5514556106 E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
8357234146 KE = \frac{1}{2} m v^2 kinetic energy Kinetic_energy
3658066792 KE_2, v_2
 
8082557839 KE, v  
7676652285 KE_2 = \frac{1}{2} m v_2^2
2790304597 KE_1, v_1
 
9880327189 KE, v  
4928007622 KE_1 = \frac{1}{2} m v_1^2
5733146966 KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
6554292307 t
4270680309 \frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
5781981178 x^2 - y^2 = (x+y)(x-y) difference of squares Difference_of_two_squares
5946759357 v_2, v_1
 
6675701064 x, y  
4648451961 v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
9356924046 \frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
9397152918 v = \frac{v_1 + v_2}{2} average velocity
2857430695 a = \frac{v_2 - v_1}{t}
acceleration
7735737409 \frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
4784793837 \frac{KE_2 - KE_1}{t} = m v a
5345738321 F = m a Newton's second law of motion Newton%27s_laws_of_motion#Newton's_second_law
2186083170 \frac{KE_2 - KE_1}{t} = v F
6715248283 PE = -F x potential energy Potential_energy
3852078149 PE_2, x_2
 
7388921188 PE, x  
2431507955 PE_2 = -F x_2
3412641898 PE_1, x_1
 
9937169749 PE, x  
4669290568 PE_1 = -F x_1
7734996511 PE_2 - PE_1 = -F ( x_2 - x_1 )
2016063530 t
7267155233 \frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
9337785146 v = \frac{x_2 - x_1}{t} average velocity
4872970974 \frac{PE_2 - PE_1}{t} = -F v
2081689540 t
2770069250 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
3591237106 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
1772416655 \frac{E_2 - E_1}{t} = v F - F v
1809909100 \frac{E_2 - E_1}{t} = 0
5778176146 t
3806977900 E_2 - E_1 = 0
5960438249 E_1
8558338742 E_2 = E_1
8945218208 \theta_{Brewster} + \theta_{refracted} = 90^{\circ} based on figure 34-27 on page 824 in 2001_HRW
9025853427 \theta_{Brewster}
1310571337 \theta_{refracted} = 90^{\circ} - \theta_{Brewster}
6450985774 n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 ) Law of Refraction eq 34-44 on page 819 in 2001_HRW
1779497048 \theta_{Brewster}, \theta_{refraction}  
7322344455 \theta_1, \theta_2
 
2575937347 n_1 \sin( \theta_{Brewster} ) = n_2 \sin( \theta_{refraction} )
7696214507 n_1 \sin( \theta_{Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{Brewster} )
8588429722 \sin( 90^{\circ} - x ) = \cos( x )
7375348852 \theta_{Brewster}
1512581563 x
6831637424 \sin( 90^{\circ} - \theta_{Brewster} ) = \cos( \theta_{Brewster} )
3061811650 n_1 \sin( \theta_{Brewster} ) = n_2 \cos( \theta_{Brewster} )
7857757625 n_1
9756089533 \sin( \theta_{Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{Brewster} )
5632428182 \cos( \theta_{Brewster} )
2768857871 \frac{\sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )} = \frac{n_2}{n_1}
4968680693 \tan( x ) = \frac{ \sin( x )}{\cos( x )}
7321695558 \theta_{Brewster}
9906920183 x
4501377629 \tan( \theta_{Brewster} ) = \frac{ \sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )}
3417126140 \tan( \theta_{Brewster} ) = \frac{ n_2 }{ n_1 }
5453995431 \arctan{ x }
6023986360 x
8495187962 \theta_{Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
9040079362 f
9565166889 T
5426308937 v = \frac{d}{t}
7476820482 C
1277713901 d
6946088325 v = \frac{C}{t}
6785303857 C = 2 \pi r
4816195267 C, r
 
2113447367 C_{\rm{Earth\ orbit}}, r_{\rm{Earth\ orbit}}
 
6348260313 C_{\rm{Earth\ orbit}} = 2 \pi r_{\rm{Earth\ orbit}}
3847777611 C, v, t  
6406487188 C_{\rm{Earth\ orbit}}, v_{\rm{Earth\ orbit}}, t_{\rm{Earth\ orbit}}
 
3046191961 v_{\rm{Earth\ orbit}} = \frac{C_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
3080027960 v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
7175416299 t_{\rm{Earth\ orbit}} = 1 {\rm{year}}
3219318145 \frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}
8721295221 t_{\rm{Earth\ orbit}} = 3.16 10^7 {\rm{seconds}}
4593428198 v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{3.16\ 10^7 {\rm seconds}}
3472836147 r_{\rm{Earth\ orbit}} = 1.496\ 10^8 {\rm km}
6998364753 v_{\rm{Earth\ orbit}} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
4180845508 v_{\rm{Earth\ orbit}} = 29.8 \frac{{\rm km}}{{\rm sec}}
4662369843 x' = \gamma (x - v t)
2983053062 x = \gamma (x' + v t')
3426941928 x = \gamma ( \gamma (x - v t) + v t' )
2096918413 x = \gamma ( \gamma x - \gamma v t + v t' )
7741202861 x = \gamma^2 x - \gamma^2 v t + \gamma v t'
7337056406 \gamma^2 x
4139999399 x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
3495403335 x
9031609275 x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
8014566709 \gamma^2 v t
9409776983 x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
2226340358 \gamma v
6417705759 \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = \gamma v t'
 
8730201316 \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'
first term was multiplied by \gamma/\gamma
1974334644 \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
5148266645 t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
4287102261 x^2 + y^2 + z^2 = c^2 t^2 describes a spherical wavefront
1201689765 x'^2 + y'^2 + z'^2 = c^2 t'^2
describes a spherical wavefront for an observer in a moving frame of reference
7057864873 y' = y
frame of reference is moving only along x direction
8515803375 z' = z
frame of reference is moving only along x direction
6153463771 3
 
4746576736 asdf
 
1027270623 minga
 
6101803533 ming
 
9231495607 a = b
 
8664264194 b + 2
 
9805063945 \gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2
4057686137 C \to C_{\rm{Earth\ orbit}}
2346150725 r \to r_{\rm{Earth\ orbit}}
9753878784 v \to v_{\rm{Earth\ orbit}}
8135396036 t \to t_{\rm{Earth\ orbit}}
2773628333 \theta_1 \to \theta_{Brewster}
7154592211 \theta_2 \to \theta_{\rm refracted}
3749492596 E \to E_1
1258245373 E \to E_2
4147101187 KE \to KE_1
3809726424 PE \to PE_1
6383056612 KE \to KE_2
5075406409 PE \to PE_2
5398681503 v \to v_1
9305761407 v \to v_2
4218009993 x \to x_1
4188639044 x \to x_2
8173074178 x \to v_2
1025759423 y \to v_1
1935543849 \gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2
1586866563 \left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)
3182633789 \gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
1916173354 -\gamma^2 v^2 + c^2 \gamma^2 = c^2
2076171250 -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0
5284610349 \gamma^2
2417941373 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
5787469164 1 - \gamma^2
1639827492 - c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
5669500954 v^2 \gamma^2
5763749235 -c^2 + c^2 \gamma^2 = v^2 \gamma^2
6408214498 c^2
2999795755 c^2 \gamma^2 = v^2 \gamma^2 + c^2
3412946408 v^2 \gamma^2
2542420160 c^2 \gamma^2 - v^2 \gamma^2 = c^2
7743841045 \gamma^2
7513513483 \gamma^2 (c^2 - v^2) = c^2
8571466509 c^2 - \gamma^2
7906112355 \gamma^2 = \frac{c^2}{c^2 - \gamma^2}
3060139639 1/2
 
1528310784 \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
8360117126 \gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}} not a physically valid result in this context
3366703541 a = \frac{v - v_0}{t} acceleration is the average change in speed over a duration
7083390553 t
4748157455 a t = v - v_0
6417359412 v_0
4798787814 a t + v_0 = v
3462972452 v = v_0 + a t
3411994811 v_{\rm ave} = \frac{d}{t}
6175547907 v_{\rm ave} = \frac{v + v_0}{2}
9897284307 \frac{d}{t} = \frac{v + v_0}{2}
8865085668 t
8706092970 d = \left(\frac{v + v_0}{2}\right)t
7011114072 d = \frac{(v_0 + a t) + v_0}{2} t
1265150401 d = \frac{2 v_0 + a t}{2} t
9658195023 d = v_0 t + \frac{1}{2} a t^2
5799753649 2
7215099603 v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
5144263777 v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
7939765107 v^2 = v_0^2 + 2 a d
6729698807 v_0
9759901995 v - v_0 = a t
2242144313 a
1967582749 t = \frac{v - v_0}{a}
5733721198 d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
5611024898 d = \frac{1}{2 a} (v^2 - v_0^2)
5542390646 2 a
8269198922 2 a d = v^2 - v_0^2
9070454719 v_0^2
4948763856 2 a d + v_0^2 = v^2
9645178657 a t
6457044853 v - a t = v_0
1259826355 d = (v - a t) t + \frac{1}{2} a t^2
4580545876 d = v t - a t^2 + \frac{1}{2} a t^2
6421241247 d = v t - \frac{1}{2} a t^2
4182362050 Z = |Z| \exp( i \theta ) Z \in \mathbb{C}
1928085940 Z^* = |Z| \exp( -i \theta )
9191880568 Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
3350830826 Z Z^* = |Z|^2
8396997949 I = | A + B |^2 intensity of two waves traveling opposite directions on same path
4437214608 Z
5623794884 A + B
2236639474 (A + B)(A + B)^* = |A + B|^2
1020854560 I = (A + B)(A + B)^*
6306552185 I = (A + B)(A^* + B^*)
8065128065 I = A A^* + B B^* + A B^* + B A^*
9761485403 Z
8710504862 A
4075539836 A A^* = |A|^2
6529120965 B
1511199318 Z
7107090465 B B^* = |B|^2
5125940051 I = |A|^2 + B B^* + A B^* + B A^*
1525861537 I = |A|^2 + |B|^2 + A B^* + B A^*
1742775076 Z \to B
7607271250 \theta \to \phi
4192519596 B = |B| \exp(i \phi)
2064205392 A
1894894315 Z
1357848476 A = |A| \exp(i \theta)
6555185548 A^* = |A| \exp(-i \theta)
4504256452 B^* = |B| \exp(-i \phi)
7621705408 I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
3085575328 I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
4935235303 x
2293352649 \theta - \phi
3660957533 \cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
3967985562 2
2700934933 2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
8497631728 I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
6774684564 \theta = \phi for coherent waves
2099546947 I = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )  
2719691582 |A| = |B| in a loop
3257276368 I = |A|^2 + |A|^2 + |A| |A| 2
 
8283354808 I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
8046208134 I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
1172039918 I_{\rm coherent} = 4 |A|^2
8602221482 \langle \cos(\theta - \phi) \rangle = 0 incoherent light source
6240206408 I_{\rm incoherent} = |A|^2 + |B|^2
6529793063 I_{\rm incoherent} = |A|^2 + |A|^2
3060393541 I_{\rm incoherent} = 2|A|^2
6556875579 \frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
8750500372 f
 
7543102525 g
 
8267133829 a = b
 
7968822469 c = d
 
3169580383 \vec{a} = \frac{d\vec{v}}{dt} acceleration is the change in speed over a duration
8602512487 \vec{a} = a_x \hat{x} + a_y \hat{y} decompose acceleration into two components
4158986868 a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
5349866551 \vec{v} = v_x \hat{x} + v_y \hat{y}
7729413831 a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
1819663717 a_x = \frac{d}{dt} v_x
8228733125 a_y = \frac{d}{dt} v_y
9707028061 a_x = 0
2741489181 a_y = -g
3270039798 2
8880467139 2
8750379055 0 = \frac{d}{dt} v_x
1977955751 -g = \frac{d}{dt} v_y
6672141531 dt
1702349646 -g dt = d v_y
8584698994 -g \int dt = \int d v_y
9973952056 -g t = v_y - v_{0, y}
4167526462 v_{0, y}
6572039835 -g t + v_{0, y} = v_y
7252338326 v_y = \frac{dy}{dt}
6204539227 -g t + v_{0, y} = \frac{dy}{dt}
1614343171 dt
8145337879 -g t dt + v_{0, y} dt = dy
8808860551 -g \int t dt + v_{0, y} \int dt = \int dy
2858549874 - \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
6098638221 y_0
2461349007 - \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
8717193282 dt
1166310428 0 dt = d v_x
2366691988 \int 0 dt = \int d v_x
1676472948 0 = v_x - v_{0, x}
1439089569 v_{0, x}
6134836751 v_{0, x} = v_x
8460820419 v_x = \frac{dx}{dt}
7455581657 v_{0, x} = \frac{dx}{dt}
8607458157 dt
1963253044 v_{0, x} dt = dx
3676159007 v_{0, x} \int dt = \int dx
9882526611 v_{0, x} t = x - x_0
3182907803 x_0
8486706976 v_{0, x} t + x_0 = x
1306360899 x = v_{0, x} t + x_0
7049769409 2
9341391925 \vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
6410818363 \theta
7391837535 \cos(\theta) = \frac{v_{0, x}}{v_0}
7376526845 \sin(\theta) = \frac{v_{0, y}}{v_0}
5868731041 v_0
6083821265 v_0 \cos(\theta) = v_{0, x}
5438722682 x = v_0 t \cos(\theta) + x_0
5620558729 v_0
8949329361 v_0 \sin(\theta) = v_{0, y}
1405465835 y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
9862900242 y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
6050070428 v_{0, x}
3274926090 t = \frac{x - x_0}{v_{0, x}}
7354529102 y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0
8406170337 y \to y_f
2403773761 t \to t_f
5379546684 y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
5373931751 t = t_f
9112191201 y_f = 0
8198310977 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
1650441634 y_0 = 0
define coordinate system such that initial height is at origin
1087417579 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
4829590294 t_f
2086924031 0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
6974054946 \frac{1}{2} g t_f
1191796961 \frac{1}{2} g t_f = v_0 \sin(\theta)
2510804451 2/g
4778077984 t_f = \frac{2 v_0 \sin(\theta)}{g}
3273630811 x \to x_f
6732786762 t \to t_f
3485125659 x_f = v_0 t_f \cos(\theta) + x_0
4370074654 t = t_f
2378095808 x_f = x_0 + d
4268085801 x_0 + d = v_0 t_f \cos(\theta) + x_0
8072682558 x_0
7233558441 d = v_0 t_f \cos(\theta)
2297105551 d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
7587034465 \theta
7214442790 x
2519058903 \sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
8922441655 d = \frac{v_0^2}{g} \sin(2 \theta)
5667870149 \theta
1541916015 \theta = \frac{\pi}{4}
3607070319 d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
5353282496 d = \frac{v_0^2}{g}
0000040490 a^2
0000999900 b/(2a)
0001030901 \cos(x)
0001111111 (\sin(x))^2
0001209482 2 \pi
0001304952 \hbar
0001334112 W
0001921933 2 i
0002239424 2
0002338514 \vec{p}_{2}
0002342425 m/m
0002367209 1/2
 
0002393922 x
0002424922 a
0002436656 i \hbar
0002449291 b/(2a)
0002838490 b/(2a)
0002919191 \sin(-x)
0002929944 1/2
0002940021 2 \pi
0003232242 t
0003413423 \cos(-x)
0003747849 -1
0003838111 2
0003919391 x
0003949052 -x
0003949921 \hbar
0003954314 dx
0003981813 -\sin(x)
0004089571 2x
0004264724 y
0004307451 (b/(2a))^2
0004582412 x
0004829194 2
0004831494 a
0004849392 x
0004858592 h
0004934845 x
0004948585 a
0005395034 a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
0005626421 t
0005749291 f
0005938585 \frac{-\hbar^2}{2m}
0006466214 (\sin(x))^2
0006544644 t
0006563727 x
0006644853 c/a
0006656532 e
0007471778 2(\sin(x))^2
0007563791 i
0007636749 x
0007894942 (\sin(x))^2
0008837284 T
0008842811 \cos(2x)
0009458842 \psi(x)
0009484724 \frac{n \pi}{W}x
0009485857 a^2\frac{2}{W}
0009485858 \frac{2n\pi}{W}
0009492929 v du
0009587738 \psi
0009594995 1/2
 
0009877781 y
0203024440 1 = \int_0^W a \sin(\frac{n \pi}{W} x) \psi(x)^* dx
0404050504 \lambda = \frac{v}{f}
0439492440 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\right|_0^W
0934990943 k = \frac{2 \pi}{v T}
0948572140 \int \cos(a x) dx = \frac{1}{a}\sin(a x)
1010393913 \langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*
1010393944 x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
1010923823 k W = n \pi
1020010291 0 = a \sin(k W)
1020394900 p = h/\lambda
1020394902 E = h f
1029039903 p = m v
1029039904 p^2 = m^2 v^2
1158485859 \frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
1202310110 \frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1202312210 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1203938249 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
1248277773 \cos(2x) = 1-2(\sin(x))^2
1293913110 0 = b
1293923844 \lambda = v T
1314464131 \vec{\nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
1314864131 \vec{\nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
1395858355 x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
1492811142 f = x - d  
1492842000 \nabla \vec{x} = f(y)  
1636453295 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = - \nabla^2 \vec{E}
1638282134 \vec{p}_{before} = \vec{p}_{after}
1648958381 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{\nabla}\psi(\vec{r},t)\right)
1857710291 0 = a \sin(n \pi)
1858578388 \nabla^2 E(\vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E(\vec{r})\exp(i \omega t)
1858772113 k = \frac{n \pi}{W}
1934748140 \int |\psi(x)|^2 dx = 1
2029293929 \nabla^2 E(\vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r})\exp(i \omega t)
2103023049 \sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right)
2113211456 f = 1/T
2123139121 -\exp(-i x) = -\cos(x)+i \sin(x)
2131616531 T f = 1
2258485859 {\cal H} \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
2394240499 x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
2394853829 \exp(-i x) = \cos(-x)+i \sin(-x)
2394935831 ( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
2394935835 \left(\langle\psi| \hat{A} |\psi\rangle \right)^+ = \left(\langle a \rangle\right)^+
2395958385 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{-p^2}{\hbar} \psi(\vec{r},t)
2404934990 \langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
2494533900 \langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
2569154141 \vec{\nabla} \times \frac{\partial}{\partial t}\vec{H} = \epsilon_0 \frac{\partial^2 }{\partial t^2}\vec{E}  
2648958382 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p}\psi(\vec{r},t)\right)
2848934890 \langle a \rangle^* = \langle a \rangle
2944838499 \psi(x) = a \sin(\frac{n \pi}{W} x)
3121234211 \frac{k}{2\pi} = \lambda
3121234212 p = \frac{h k}{2\pi}
3121513111 k = \frac{2 \pi}{\lambda}
3131111133 T = 1 / f
3131211131 \omega = 2 \pi f
3132131132 \omega = \frac{2\pi}{T}
3147472131 \frac{\omega}{2 \pi} = f
3285732911 (\cos(x))^2 = 1-(\sin(x))^2
3485475729 \nabla^2 E(\vec{r}) = - \frac{\omega^2}{c^2} E(\vec{r})
3585845894 \langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
3829492824 \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right) = \cos(x)
3924948349 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
3942849294 \exp(i x)-\exp(-i x) = 2 i \sin(x)
3943939590 x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
3947269979 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
3948571256 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi(\vec{r},t)
3948572230 \vec{\nabla}\psi(\vec{r},t) = \psi_0 \vec{\nabla}\exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)  
3948574224 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\vec{k}\cdot\vec{r} - \omega t \right) \right)
3948574226 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
3948574228 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3948574230 \psi(\vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
3948574233 \frac{\partial}{\partial t}\psi(\vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3948574235 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)  
3951205425 \vec{p}_{after} = \vec{p}_{1}
4147472132 E = \frac{h \omega}{2 \pi}
4298359835 E = \frac{1}{2}m v^2
4298359845 E = \frac{1}{2m}m^2 v^2
4298359851 E = \frac{p^2}{2m}
4341171256 i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{p^2}{2 m} \psi(\vec{r},t)
4348571256 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi(\vec{r},t)
4394958389 \vec{\nabla}\cdot \left(\vec{\nabla}\psi(\vec{r},t)\right) = \frac{i}{\hbar} \vec{\nabla}\cdot\left( \vec{p} \psi(\vec{r},t)\right)
4585828572 \epsilon_0 \mu_0 = \frac{1}{c^2}
4585932229 \cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
4598294821 \exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
4638429483 \exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
4742644828 \exp(i x)+\exp(-i x) = 2 \cos(x)
4827492911 \cos(2x)+(\sin(x))^2 = 1-(\sin(x))^2
4838429483 \exp(2 i x) = \cos(2 x)+i \sin(2 x)
4843995999 \frac{1}{2 i}\left(\exp(i x)-\exp(-i x)\right) = \sin(x)
4857472413 1 = \int \psi(x)\psi(x)^* dx
4857475848 \frac{1}{a^2} = \frac{W}{2}
4858429483 \exp(i x)\exp(i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))  
4923339482 i x = \log(y)
4928239482 \log(y) = i x
4928923942 a = b  
4929218492 a+b = c  
4929482992 b = 2
 
4938429482 \exp(-i x) = \cos(x)+i \sin(-x)
4938429483 \exp(i x) = \cos(x)+i \sin(x)
4938429484 \exp(-i x) = \cos(x)-i \sin(x)
4943571230 \vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
4948934890 \langle \psi| \hat{A} |\psi\rangle = \langle a \rangle^*
4949359835 \langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
4954839242 \cos(2x)+i\sin(2x) = (\cos(x)+i \sin(x))(\cos(x)+i \sin(x))  
4978429483 \exp(-i x) = \cos(-x)+i \sin(-x)  
4984892984 a+2 = c
 
4985825552 \nabla^2 E(\vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E(\vec{r})\exp(i \omega t)
5530148480 \vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
5727578862 \frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
5832984291 (\sin(x))^2 + (\cos(x))^2 = 1
5857434758 \int a dx = a x
5868688585 \frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = \frac{p^2}{2m} \psi(\vec{r},t)
5900595848 k = \frac{\omega}{v}
5928285821 x^2 + 2x(b/(2a)) + (b/(2a))^2 = (x+(b/(2a)))^2
5928292841 x^2 + (b/a)x + (b/(2a))^2 = -c/a + (b/(2a))^2
5938459282 x^2 + (b/a)x = -c/a
5958392859 x^2 + (b/a)x+(c/a) = 0
5959282914 x^2 + x(b/a) + (b/(2a))^2 = (x+(b/(2a)))^2
5982958248 x = -\sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
5982958249 x+(b/(2a)) = -\sqrt{(b/(2a))^2 - (c/a)}
5985371230 \vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi(\vec{r},t)
6742123016 \vec{p}_{electron}\cdot\vec{p}_{electron} = (\vec{p}_{1}\cdot\vec{p}_{1})+(\vec{p}_{2}\cdot\vec{p}_{2})-2(\vec{p}_{1}\cdot\vec{p}_{2})
7466829492 \vec{\nabla} \cdot \vec{E} = 0
7564894985 \int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
7572664728 \cos(2x)+2(\sin(x))^2 = 1
7575738420 \left(\sin\left(\frac{n \pi}{W}x\right)\right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
7575859295 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859300 \epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{\nabla} \times \vec{E} )_k = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859302 \epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859304 \epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
7575859306 \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859308 \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859310 \hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859312 \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7917051060 \vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
8139187332 \vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
8257621077 \vec{p}_{before} = \vec{p}_{1}
8311458118 \vec{p}_{after} = \vec{p}_{2}+\vec{p}_{electron}
8399484849 \langle x^2 -2x\langle x \rangle+\langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
8484544728 -a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
8485757728 a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
8485867742 \frac{2}{W} = a^2
8489593958 d(u v) = u dv + v du
8489593960 d(u v) - v du = u dv
8489593962 u dv = d(u v) - v du
8489593964 \int u dv = u v - \int v du
8494839423 \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
8572657110 1 = \int |\psi(x)|^2 dx
8572852424 \vec{E} = E(\vec{r},t)
8575746378 \int \frac{1}{2} dx = \frac{1}{2} x
8575748999 \frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x)\right) = -k^2 \left(a \sin(kx) + b \cos(kx)\right)
8576785890 1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
8577275751 0 = a \sin(0) + b\cos(0)
8582885111 \psi(x) = a \sin(kx) + b \cos(kx)
8582954722 x^2 + 2xh + h^2 = (x+h)^2
8849289982 \psi(x)^* = a \sin(\frac{n \pi}{W} x)
8889444440 1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right)\right)^2 dx
9059289981 \psi(x) = a \sin(k x)
9285928292 ax^2 + bx + c = 0
9291999979 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t}
9294858532 \hat{A}^+ = \hat{A}
9385938295 (x+(b/(2a)))^2 = -(c/a) + (b/(2a))^2
9393939991 \psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9393939992 \psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9394939493 \nabla^2 E(\vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r},t)
9429829482 \frac{d}{dx} y = -\sin(x) + i\cos(x)
9482113948 \frac{dy}{y} = i dx
9482438243 (\cos(x))^2 = \cos(2x)+(\sin(x))^2
9482923849 \exp(i x) = y
9482928242 \cos(2x) = (\cos(x))^2 - (\sin(x))^2
9482928243 \cos(2x)+(\sin(x))^2 = (\cos(x))^2
9482943948 \log(y) = i dx
9482948292 b = c
 
9482984922 \frac{d}{dx} y = (i\sin(x) + \cos(x)) i
9483928192 \cos(2x)+i\sin(2x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
9485384858 \nabla^2 E(\vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E(\vec{r})\exp(i \omega t)
9485747245 \sqrt{\frac{2}{W}} = a
9485747246 -\sqrt{\frac{2}{W}} = a
9492920340 y = \cos(x)+i \sin(x)
9494829190 k  
9495857278 \psi(x=W) = 0
9499428242 E(\vec{r},t) = E(\vec{r})\exp(i \omega t)
9499959299 a + k = b + k  
9582958293 x = \sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
9582958294 x+(b/(2a)) = \sqrt{(b/(2a))^2 - (c/a)}
9584821911 c + d = a  
9585727710 \psi(x=0) = 0
9596004948 x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
9848292229 dy = y i dx
9848294829 \frac{d}{dx} y = y i
9858028950 \frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
9889984281 2(\sin(x))^2 = 1-\cos(2x)
9919999981 \rho = 0
9958485859 \frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
9988949211 (\sin(x))^2 = \frac{1-\cos(2x)}{2}
9991999979 \vec{\nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
9999998870 \frac{\vec{p}}{\hbar} = \vec{k}
9999999870 \frac{p}{\hbar} = k
9999999950 \beta = 1/(k_{Boltzmann}T)  
9999999951 \langle x | k \rangle = \frac{\exp(i k x)}{\sqrt{2\pi}}  
9999999952 f(x) \delta(x-a) = f(a)
 
9999999953 \int_{-\infty}^{\infty} \delta(x) dx = 1
 
9999999954 c = 1/\sqrt{\epsilon_0 \mu_0}  
9999999955 \vec{E} = -\vec{\nabla} \Psi
 
9999999956 \vec{F} = \frac{\partial}{\partial t}\vec{p}  
9999999957 \vec{F} = -\vec{\nabla} V
 
9999999960 \hbar = h/(2 \pi)
9999999961 \frac{E}{\hbar} = \omega
9999999962 p = \hbar k
9999999963 \lambda = h/p
 
9999999964 \omega = c k  
9999999965 E = \omega \hbar
9999999966 \vec{L} = \vec{r}\times\vec{p}
 
9999999967 \vec{S} = \frac{1}{\mu_0} \vec{E}\times \vec{B}
 
9999999968 x = \frac{-b-\sqrt{b^2-4ac}}{2a}
9999999969 x = \frac{-b+\sqrt{b^2-4ac}}{2a}
9999999970 \eta_1 \sin\theta_1 = \eta_2 \sin\theta_2
 
9999999971 {\cal H} = \frac{p^2}{2m}+V
 
9999999972 {\cal H} |\psi\rangle = E |\psi\rangle
 
9999999973 \left( \Delta A \right)^2 = \langle A^2 \rangle - \langle A \rangle^2
 
9999999974 \langle \psi| \hat{A} |\psi\rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx
 
9999999975 \langle \psi| \hat{A} |\psi\rangle = \langle a \rangle
9999999976 \hat{p} = -i \hbar \frac{\partial }{\partial x}  
9999999977 [\hat{x},\hat{p}] = i \hbar  
9999999978 \vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}
 
9999999979 \vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}  
9999999980 \vec{\nabla} \cdot \vec{B} = 0
 
9999999981 \vec{\nabla} \cdot \vec{E} = \rho/\epsilon_0
9999999982 V = I R + Q/C + L \frac{\partial I}{\partial t}  
9999999983 C = V A/d
 
9999999984 Q = C V
 
9999999985 V = I R
 
9999999986 \left[cos⁡θsin⁡θ−sin⁡θcos⁡θ\right]\left[AxAy\right] = \left[Ax′Ay′\right]
 
9999999987 1 atmosphere = 101.325 Pascal
 
9999999988 1 atmosphere = 14.7 pounds/(inch^2)
 
9999999989 mass_{electron} = 511000 electronVolts/(q^2)
 
9999999990 Tesla = 10000*Gauss
 
9999999991 Pascal = Newton / (meter^2)
 
9999999992 Tesla = Newton / (Ampere*meter)
 
9999999993 Farad = Coulumb / Volt
 
9999999995 Volt = Joule / Coulumb
 
9999999996 Coulomb = Ampere / second
 
9999999997 Watt = Joule / second
 
9999999998 Joule = Newton*meter
 
9999999999 Newton = kilogram*meter/(second^2)
 
4961662865 x
9110536742 2 x
8483686863 \sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x)\right)
3470587782 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
8642992037 2
9894826550 2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \left(\exp(i x)+\exp(-i x)\right)
4257214632 \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)  
8699789241 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
9180861128 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
2405307372 \sin(2 x) = 2 \sin(x) \cos(x)
3268645065 x
9350663581 \pi
8332931442 \exp(i \pi) = \cos(\pi)+i \sin(\pi)
6885625907 \exp(i \pi) = -1 + i 0
3331824625 \exp(i \pi) = -1
4901237716 1
2501591100 \exp(i \pi) + 1 = 0
5136652623 E = KE + PE mechanical energy is the sum of the potential plus kinetic energies
7915321076 E, KE, PE  
3192374582 E_2, KE_2, PE_2
 
7875206161 E_2 = KE_2 + PE_2
4229092186 E, KE, PE  
1490821948 E_1, KE_1, PE_1
 
4303372136 E_1 = KE_1 + PE_1
5514556106 E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
8357234146 KE = \frac{1}{2} m v^2 kinetic energy Kinetic_energy
3658066792 KE_2, v_2
 
8082557839 KE, v  
7676652285 KE_2 = \frac{1}{2} m v_2^2
2790304597 KE_1, v_1
 
9880327189 KE, v  
4928007622 KE_1 = \frac{1}{2} m v_1^2
5733146966 KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
6554292307 t
4270680309 \frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
5781981178 x^2 - y^2 = (x+y)(x-y) difference of squares Difference_of_two_squares
5946759357 v_2, v_1
 
6675701064 x, y  
4648451961 v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
9356924046 \frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
9397152918 v = \frac{v_1 + v_2}{2} average velocity
2857430695 a = \frac{v_2 - v_1}{t}
acceleration
7735737409 \frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
4784793837 \frac{KE_2 - KE_1}{t} = m v a
5345738321 F = m a Newton's second law of motion Newton%27s_laws_of_motion#Newton's_second_law
2186083170 \frac{KE_2 - KE_1}{t} = v F
6715248283 PE = -F x potential energy Potential_energy
3852078149 PE_2, x_2
 
7388921188 PE, x  
2431507955 PE_2 = -F x_2
3412641898 PE_1, x_1
 
9937169749 PE, x  
4669290568 PE_1 = -F x_1
7734996511 PE_2 - PE_1 = -F ( x_2 - x_1 )
2016063530 t
7267155233 \frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
9337785146 v = \frac{x_2 - x_1}{t} average velocity
4872970974 \frac{PE_2 - PE_1}{t} = -F v
2081689540 t
2770069250 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
3591237106 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
1772416655 \frac{E_2 - E_1}{t} = v F - F v
1809909100 \frac{E_2 - E_1}{t} = 0
5778176146 t
3806977900 E_2 - E_1 = 0
5960438249 E_1
8558338742 E_2 = E_1
8945218208 \theta_{Brewster} + \theta_{refracted} = 90^{\circ} based on figure 34-27 on page 824 in 2001_HRW
9025853427 \theta_{Brewster}
1310571337 \theta_{refracted} = 90^{\circ} - \theta_{Brewster}
6450985774 n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 ) Law of Refraction eq 34-44 on page 819 in 2001_HRW
1779497048 \theta_{Brewster}, \theta_{refraction}  
7322344455 \theta_1, \theta_2
 
2575937347 n_1 \sin( \theta_{Brewster} ) = n_2 \sin( \theta_{refraction} )
7696214507 n_1 \sin( \theta_{Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{Brewster} )
8588429722 \sin( 90^{\circ} - x ) = \cos( x )
7375348852 \theta_{Brewster}
1512581563 x
6831637424 \sin( 90^{\circ} - \theta_{Brewster} ) = \cos( \theta_{Brewster} )
3061811650 n_1 \sin( \theta_{Brewster} ) = n_2 \cos( \theta_{Brewster} )
7857757625 n_1
9756089533 \sin( \theta_{Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{Brewster} )
5632428182 \cos( \theta_{Brewster} )
2768857871 \frac{\sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )} = \frac{n_2}{n_1}
4968680693 \tan( x ) = \frac{ \sin( x )}{\cos( x )}
7321695558 \theta_{Brewster}
9906920183 x
4501377629 \tan( \theta_{Brewster} ) = \frac{ \sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )}
3417126140 \tan( \theta_{Brewster} ) = \frac{ n_2 }{ n_1 }
5453995431 \arctan{ x }
6023986360 x
8495187962 \theta_{Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
9040079362 f
9565166889 T
5426308937 v = \frac{d}{t}
7476820482 C
1277713901 d
6946088325 v = \frac{C}{t}
6785303857 C = 2 \pi r
4816195267 C, r
 
2113447367 C_{\rm{Earth\ orbit}}, r_{\rm{Earth\ orbit}}
 
6348260313 C_{\rm{Earth\ orbit}} = 2 \pi r_{\rm{Earth\ orbit}}
3847777611 C, v, t  
6406487188 C_{\rm{Earth\ orbit}}, v_{\rm{Earth\ orbit}}, t_{\rm{Earth\ orbit}}
 
3046191961 v_{\rm{Earth\ orbit}} = \frac{C_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
3080027960 v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
7175416299 t_{\rm{Earth\ orbit}} = 1 {\rm{year}}
3219318145 \frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}
8721295221 t_{\rm{Earth\ orbit}} = 3.16 10^7 {\rm{seconds}}
4593428198 v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{3.16\ 10^7 {\rm seconds}}
3472836147 r_{\rm{Earth\ orbit}} = 1.496\ 10^8 {\rm km}
6998364753 v_{\rm{Earth\ orbit}} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
4180845508 v_{\rm{Earth\ orbit}} = 29.8 \frac{{\rm km}}{{\rm sec}}
4662369843 x' = \gamma (x - v t)
2983053062 x = \gamma (x' + v t')
3426941928 x = \gamma ( \gamma (x - v t) + v t' )
2096918413 x = \gamma ( \gamma x - \gamma v t + v t' )
7741202861 x = \gamma^2 x - \gamma^2 v t + \gamma v t'
7337056406 \gamma^2 x
4139999399 x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
3495403335 x
9031609275 x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
8014566709 \gamma^2 v t
9409776983 x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
2226340358 \gamma v
6417705759 \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = \gamma v t'
 
8730201316 \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'
first term was multiplied by \gamma/\gamma
1974334644 \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
5148266645 t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
4287102261 x^2 + y^2 + z^2 = c^2 t^2 describes a spherical wavefront
1201689765 x'^2 + y'^2 + z'^2 = c^2 t'^2
describes a spherical wavefront for an observer in a moving frame of reference
7057864873 y' = y
frame of reference is moving only along x direction
8515803375 z' = z
frame of reference is moving only along x direction
6153463771 3
 
4746576736 asdf
 
1027270623 minga
 
6101803533 ming
 
9231495607 a = b
 
8664264194 b + 2
 
9805063945 \gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2
4057686137 C \to C_{\rm{Earth\ orbit}}
2346150725 r \to r_{\rm{Earth\ orbit}}
9753878784 v \to v_{\rm{Earth\ orbit}}
8135396036 t \to t_{\rm{Earth\ orbit}}
2773628333 \theta_1 \to \theta_{Brewster}
7154592211 \theta_2 \to \theta_{\rm refracted}
3749492596 E \to E_1
1258245373 E \to E_2
4147101187 KE \to KE_1
3809726424 PE \to PE_1
6383056612 KE \to KE_2
5075406409 PE \to PE_2
5398681503 v \to v_1
9305761407 v \to v_2
4218009993 x \to x_1
4188639044 x \to x_2
8173074178 x \to v_2
1025759423 y \to v_1
1935543849 \gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2
1586866563 \left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)
3182633789 \gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
1916173354 -\gamma^2 v^2 + c^2 \gamma^2 = c^2
2076171250 -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0
5284610349 \gamma^2
2417941373 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
5787469164 1 - \gamma^2
1639827492 - c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
5669500954 v^2 \gamma^2
5763749235 -c^2 + c^2 \gamma^2 = v^2 \gamma^2
6408214498 c^2
2999795755 c^2 \gamma^2 = v^2 \gamma^2 + c^2
3412946408 v^2 \gamma^2
2542420160 c^2 \gamma^2 - v^2 \gamma^2 = c^2
7743841045 \gamma^2
7513513483 \gamma^2 (c^2 - v^2) = c^2
8571466509 c^2 - \gamma^2
7906112355 \gamma^2 = \frac{c^2}{c^2 - \gamma^2}
3060139639 1/2
 
1528310784 \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
8360117126 \gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}} not a physically valid result in this context
3366703541 a = \frac{v - v_0}{t} acceleration is the average change in speed over a duration
7083390553 t
4748157455 a t = v - v_0
6417359412 v_0
4798787814 a t + v_0 = v
3462972452 v = v_0 + a t
3411994811 v_{\rm ave} = \frac{d}{t}
6175547907 v_{\rm ave} = \frac{v + v_0}{2}
9897284307 \frac{d}{t} = \frac{v + v_0}{2}
8865085668 t
8706092970 d = \left(\frac{v + v_0}{2}\right)t
7011114072 d = \frac{(v_0 + a t) + v_0}{2} t
1265150401 d = \frac{2 v_0 + a t}{2} t
9658195023 d = v_0 t + \frac{1}{2} a t^2
5799753649 2
7215099603 v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
5144263777 v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
7939765107 v^2 = v_0^2 + 2 a d
6729698807 v_0
9759901995 v - v_0 = a t
2242144313 a
1967582749 t = \frac{v - v_0}{a}
5733721198 d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
5611024898 d = \frac{1}{2 a} (v^2 - v_0^2)
5542390646 2 a
8269198922 2 a d = v^2 - v_0^2
9070454719 v_0^2
4948763856 2 a d + v_0^2 = v^2
9645178657 a t
6457044853 v - a t = v_0
1259826355 d = (v - a t) t + \frac{1}{2} a t^2
4580545876 d = v t - a t^2 + \frac{1}{2} a t^2
6421241247 d = v t - \frac{1}{2} a t^2
4182362050 Z = |Z| \exp( i \theta ) Z \in \mathbb{C}
1928085940 Z^* = |Z| \exp( -i \theta )
9191880568 Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
3350830826 Z Z^* = |Z|^2
8396997949 I = | A + B |^2 intensity of two waves traveling opposite directions on same path
4437214608 Z
5623794884 A + B
2236639474 (A + B)(A + B)^* = |A + B|^2
1020854560 I = (A + B)(A + B)^*
6306552185 I = (A + B)(A^* + B^*)
8065128065 I = A A^* + B B^* + A B^* + B A^*
9761485403 Z
8710504862 A
4075539836 A A^* = |A|^2
6529120965 B
1511199318 Z
7107090465 B B^* = |B|^2
5125940051 I = |A|^2 + B B^* + A B^* + B A^*
1525861537 I = |A|^2 + |B|^2 + A B^* + B A^*
1742775076 Z \to B
7607271250 \theta \to \phi
4192519596 B = |B| \exp(i \phi)
2064205392 A
1894894315 Z
1357848476 A = |A| \exp(i \theta)
6555185548 A^* = |A| \exp(-i \theta)
4504256452 B^* = |B| \exp(-i \phi)
7621705408 I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
3085575328 I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
4935235303 x
2293352649 \theta - \phi
3660957533 \cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
3967985562 2
2700934933 2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
8497631728 I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
6774684564 \theta = \phi for coherent waves
2099546947 I = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )  
2719691582 |A| = |B| in a loop
3257276368 I = |A|^2 + |A|^2 + |A| |A| 2
 
8283354808 I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
8046208134 I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
1172039918 I_{\rm coherent} = 4 |A|^2
8602221482 \langle \cos(\theta - \phi) \rangle = 0 incoherent light source
6240206408 I_{\rm incoherent} = |A|^2 + |B|^2
6529793063 I_{\rm incoherent} = |A|^2 + |A|^2
3060393541 I_{\rm incoherent} = 2|A|^2
6556875579 \frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
8750500372 f
 
7543102525 g
 
8267133829 a = b
 
7968822469 c = d
 
3169580383 \vec{a} = \frac{d\vec{v}}{dt} acceleration is the change in speed over a duration
8602512487 \vec{a} = a_x \hat{x} + a_y \hat{y} decompose acceleration into two components
4158986868 a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
5349866551 \vec{v} = v_x \hat{x} + v_y \hat{y}
7729413831 a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
1819663717 a_x = \frac{d}{dt} v_x
8228733125 a_y = \frac{d}{dt} v_y
9707028061 a_x = 0
2741489181 a_y = -g
3270039798 2
8880467139 2
8750379055 0 = \frac{d}{dt} v_x
1977955751 -g = \frac{d}{dt} v_y
6672141531 dt
1702349646 -g dt = d v_y
8584698994 -g \int dt = \int d v_y
9973952056 -g t = v_y - v_{0, y}
4167526462 v_{0, y}
6572039835 -g t + v_{0, y} = v_y
7252338326 v_y = \frac{dy}{dt}
6204539227 -g t + v_{0, y} = \frac{dy}{dt}
1614343171 dt
8145337879 -g t dt + v_{0, y} dt = dy
8808860551 -g \int t dt + v_{0, y} \int dt = \int dy
2858549874 - \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
6098638221 y_0
2461349007 - \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
8717193282 dt
1166310428 0 dt = d v_x
2366691988 \int 0 dt = \int d v_x
1676472948 0 = v_x - v_{0, x}
1439089569 v_{0, x}
6134836751 v_{0, x} = v_x
8460820419 v_x = \frac{dx}{dt}
7455581657 v_{0, x} = \frac{dx}{dt}
8607458157 dt
1963253044 v_{0, x} dt = dx
3676159007 v_{0, x} \int dt = \int dx
9882526611 v_{0, x} t = x - x_0
3182907803 x_0
8486706976 v_{0, x} t + x_0 = x
1306360899 x = v_{0, x} t + x_0
7049769409 2
9341391925 \vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
6410818363 \theta
7391837535 \cos(\theta) = \frac{v_{0, x}}{v_0}
7376526845 \sin(\theta) = \frac{v_{0, y}}{v_0}
5868731041 v_0
6083821265 v_0 \cos(\theta) = v_{0, x}
5438722682 x = v_0 t \cos(\theta) + x_0
5620558729 v_0
8949329361 v_0 \sin(\theta) = v_{0, y}
1405465835 y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
9862900242 y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
6050070428 v_{0, x}
3274926090 t = \frac{x - x_0}{v_{0, x}}
7354529102 y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0
8406170337 y \to y_f
2403773761 t \to t_f
5379546684 y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
5373931751 t = t_f
9112191201 y_f = 0
8198310977 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
1650441634 y_0 = 0
define coordinate system such that initial height is at origin
1087417579 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
4829590294 t_f
2086924031 0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
6974054946 \frac{1}{2} g t_f
1191796961 \frac{1}{2} g t_f = v_0 \sin(\theta)
2510804451 2/g
4778077984 t_f = \frac{2 v_0 \sin(\theta)}{g}
3273630811 x \to x_f
6732786762 t \to t_f
3485125659 x_f = v_0 t_f \cos(\theta) + x_0
4370074654 t = t_f
2378095808 x_f = x_0 + d
4268085801 x_0 + d = v_0 t_f \cos(\theta) + x_0
8072682558 x_0
7233558441 d = v_0 t_f \cos(\theta)
2297105551 d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
7587034465 \theta
7214442790 x
2519058903 \sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
8922441655 d = \frac{v_0^2}{g} \sin(2 \theta)
5667870149 \theta
1541916015 \theta = \frac{\pi}{4}
3607070319 d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
5353282496 d = \frac{v_0^2}{g}
0000040490 a^2
0000999900 b/(2a)
0001030901 \cos(x)
0001111111 (\sin(x))^2
0001209482 2 \pi
0001304952 \hbar
0001334112 W
0001921933 2 i
0002239424 2
0002338514 \vec{p}_{2}
0002342425 m/m
0002367209 1/2
 
0002393922 x
0002424922 a
0002436656 i \hbar
0002449291 b/(2a)
0002838490 b/(2a)
0002919191 \sin(-x)
0002929944 1/2
0002940021 2 \pi
0003232242 t
0003413423 \cos(-x)
0003747849 -1
0003838111 2
0003919391 x
0003949052 -x
0003949921 \hbar
0003954314 dx
0003981813 -\sin(x)
0004089571 2x
0004264724 y
0004307451 (b/(2a))^2
0004582412 x
0004829194 2
0004831494 a
0004849392 x
0004858592 h
0004934845 x
0004948585 a
0005395034 a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
0005626421 t
0005749291 f
0005938585 \frac{-\hbar^2}{2m}
0006466214 (\sin(x))^2
0006544644 t
0006563727 x
0006644853 c/a
0006656532 e
0007471778 2(\sin(x))^2
0007563791 i
0007636749 x
0007894942 (\sin(x))^2
0008837284 T
0008842811 \cos(2x)
0009458842 \psi(x)
0009484724 \frac{n \pi}{W}x
0009485857 a^2\frac{2}{W}
0009485858 \frac{2n\pi}{W}
0009492929 v du
0009587738 \psi
0009594995 1/2
 
0009877781 y
0203024440 1 = \int_0^W a \sin(\frac{n \pi}{W} x) \psi(x)^* dx
0404050504 \lambda = \frac{v}{f}
0439492440 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\right|_0^W
0934990943 k = \frac{2 \pi}{v T}
0948572140 \int \cos(a x) dx = \frac{1}{a}\sin(a x)
1010393913 \langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*
1010393944 x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
1010923823 k W = n \pi
1020010291 0 = a \sin(k W)
1020394900 p = h/\lambda
1020394902 E = h f
1029039903 p = m v
1029039904 p^2 = m^2 v^2
1158485859 \frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
1202310110 \frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1202312210 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1203938249 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
1248277773 \cos(2x) = 1-2(\sin(x))^2
1293913110 0 = b
1293923844 \lambda = v T
1314464131 \vec{\nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
1314864131 \vec{\nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
1395858355 x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
1492811142 f = x - d  
1492842000 \nabla \vec{x} = f(y)  
1636453295 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = - \nabla^2 \vec{E}
1638282134 \vec{p}_{before} = \vec{p}_{after}
1648958381 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{\nabla}\psi(\vec{r},t)\right)
1857710291 0 = a \sin(n \pi)
1858578388 \nabla^2 E(\vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E(\vec{r})\exp(i \omega t)
1858772113 k = \frac{n \pi}{W}
1934748140 \int |\psi(x)|^2 dx = 1
2029293929 \nabla^2 E(\vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r})\exp(i \omega t)
2103023049 \sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right)
2113211456 f = 1/T
2123139121 -\exp(-i x) = -\cos(x)+i \sin(x)
2131616531 T f = 1
2258485859 {\cal H} \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
2394240499 x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
2394853829 \exp(-i x) = \cos(-x)+i \sin(-x)
2394935831 ( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
2394935835 \left(\langle\psi| \hat{A} |\psi\rangle \right)^+ = \left(\langle a \rangle\right)^+
2395958385 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{-p^2}{\hbar} \psi(\vec{r},t)
2404934990 \langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
2494533900 \langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
2569154141 \vec{\nabla} \times \frac{\partial}{\partial t}\vec{H} = \epsilon_0 \frac{\partial^2 }{\partial t^2}\vec{E}  
2648958382 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p}\psi(\vec{r},t)\right)
2848934890 \langle a \rangle^* = \langle a \rangle
2944838499 \psi(x) = a \sin(\frac{n \pi}{W} x)
3121234211 \frac{k}{2\pi} = \lambda
3121234212 p = \frac{h k}{2\pi}
3121513111 k = \frac{2 \pi}{\lambda}
3131111133 T = 1 / f
3131211131 \omega = 2 \pi f
3132131132 \omega = \frac{2\pi}{T}
3147472131 \frac{\omega}{2 \pi} = f
3285732911 (\cos(x))^2 = 1-(\sin(x))^2
3485475729 \nabla^2 E(\vec{r}) = - \frac{\omega^2}{c^2} E(\vec{r})
3585845894 \langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
3829492824 \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right) = \cos(x)
3924948349 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
3942849294 \exp(i x)-\exp(-i x) = 2 i \sin(x)
3943939590 x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
3947269979 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
3948571256 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi(\vec{r},t)
3948572230 \vec{\nabla}\psi(\vec{r},t) = \psi_0 \vec{\nabla}\exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)  
3948574224 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\vec{k}\cdot\vec{r} - \omega t \right) \right)
3948574226 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
3948574228 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3948574230 \psi(\vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
3948574233 \frac{\partial}{\partial t}\psi(\vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3948574235 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)  
3951205425 \vec{p}_{after} = \vec{p}_{1}
4147472132 E = \frac{h \omega}{2 \pi}
4298359835 E = \frac{1}{2}m v^2
4298359845 E = \frac{1}{2m}m^2 v^2
4298359851 E = \frac{p^2}{2m}
4341171256 i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{p^2}{2 m} \psi(\vec{r},t)
4348571256 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi(\vec{r},t)
4394958389 \vec{\nabla}\cdot \left(\vec{\nabla}\psi(\vec{r},t)\right) = \frac{i}{\hbar} \vec{\nabla}\cdot\left( \vec{p} \psi(\vec{r},t)\right)
4585828572 \epsilon_0 \mu_0 = \frac{1}{c^2}
4585932229 \cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
4598294821 \exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
4638429483 \exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
4742644828 \exp(i x)+\exp(-i x) = 2 \cos(x)
4827492911 \cos(2x)+(\sin(x))^2 = 1-(\sin(x))^2
4838429483 \exp(2 i x) = \cos(2 x)+i \sin(2 x)
4843995999 \frac{1}{2 i}\left(\exp(i x)-\exp(-i x)\right) = \sin(x)
4857472413 1 = \int \psi(x)\psi(x)^* dx
4857475848 \frac{1}{a^2} = \frac{W}{2}
4858429483 \exp(i x)\exp(i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))  
4923339482 i x = \log(y)
4928239482 \log(y) = i x
4928923942 a = b  
4929218492 a+b = c  
4929482992 b = 2
 
4938429482 \exp(-i x) = \cos(x)+i \sin(-x)
4938429483 \exp(i x) = \cos(x)+i \sin(x)
4938429484 \exp(-i x) = \cos(x)-i \sin(x)
4943571230 \vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
4948934890 \langle \psi| \hat{A} |\psi\rangle = \langle a \rangle^*
4949359835 \langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
4954839242 \cos(2x)+i\sin(2x) = (\cos(x)+i \sin(x))(\cos(x)+i \sin(x))  
4978429483 \exp(-i x) = \cos(-x)+i \sin(-x)  
4984892984 a+2 = c
 
4985825552 \nabla^2 E(\vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E(\vec{r})\exp(i \omega t)
5530148480 \vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
5727578862 \frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
5832984291 (\sin(x))^2 + (\cos(x))^2 = 1
5857434758 \int a dx = a x
5868688585 \frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = \frac{p^2}{2m} \psi(\vec{r},t)
5900595848 k = \frac{\omega}{v}
5928285821 x^2 + 2x(b/(2a)) + (b/(2a))^2 = (x+(b/(2a)))^2
5928292841 x^2 + (b/a)x + (b/(2a))^2 = -c/a + (b/(2a))^2
5938459282 x^2 + (b/a)x = -c/a
5958392859 x^2 + (b/a)x+(c/a) = 0
5959282914 x^2 + x(b/a) + (b/(2a))^2 = (x+(b/(2a)))^2
5982958248 x = -\sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
5982958249 x+(b/(2a)) = -\sqrt{(b/(2a))^2 - (c/a)}
5985371230 \vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi(\vec{r},t)
6742123016 \vec{p}_{electron}\cdot\vec{p}_{electron} = (\vec{p}_{1}\cdot\vec{p}_{1})+(\vec{p}_{2}\cdot\vec{p}_{2})-2(\vec{p}_{1}\cdot\vec{p}_{2})
7466829492 \vec{\nabla} \cdot \vec{E} = 0
7564894985 \int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
7572664728 \cos(2x)+2(\sin(x))^2 = 1
7575738420 \left(\sin\left(\frac{n \pi}{W}x\right)\right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
7575859295 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859300 \epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{\nabla} \times \vec{E} )_k = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859302 \epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859304 \epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
7575859306 \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859308 \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859310 \hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859312 \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7917051060 \vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
8139187332 \vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
8257621077 \vec{p}_{before} = \vec{p}_{1}
8311458118 \vec{p}_{after} = \vec{p}_{2}+\vec{p}_{electron}
8399484849 \langle x^2 -2x\langle x \rangle+\langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
8484544728 -a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
8485757728 a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
8485867742 \frac{2}{W} = a^2
8489593958 d(u v) = u dv + v du
8489593960 d(u v) - v du = u dv
8489593962 u dv = d(u v) - v du
8489593964 \int u dv = u v - \int v du
8494839423 \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
8572657110 1 = \int |\psi(x)|^2 dx
8572852424 \vec{E} = E(\vec{r},t)
8575746378 \int \frac{1}{2} dx = \frac{1}{2} x
8575748999 \frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x)\right) = -k^2 \left(a \sin(kx) + b \cos(kx)\right)
8576785890 1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
8577275751 0 = a \sin(0) + b\cos(0)
8582885111 \psi(x) = a \sin(kx) + b \cos(kx)
8582954722 x^2 + 2xh + h^2 = (x+h)^2
8849289982 \psi(x)^* = a \sin(\frac{n \pi}{W} x)
8889444440 1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right)\right)^2 dx
9059289981 \psi(x) = a \sin(k x)
9285928292 ax^2 + bx + c = 0
9291999979 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t}
9294858532 \hat{A}^+ = \hat{A}
9385938295 (x+(b/(2a)))^2 = -(c/a) + (b/(2a))^2
9393939991 \psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9393939992 \psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9394939493 \nabla^2 E(\vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r},t)
9429829482 \frac{d}{dx} y = -\sin(x) + i\cos(x)
9482113948 \frac{dy}{y} = i dx
9482438243 (\cos(x))^2 = \cos(2x)+(\sin(x))^2
9482923849 \exp(i x) = y
9482928242 \cos(2x) = (\cos(x))^2 - (\sin(x))^2
9482928243 \cos(2x)+(\sin(x))^2 = (\cos(x))^2
9482943948 \log(y) = i dx
9482948292 b = c
 
9482984922 \frac{d}{dx} y = (i\sin(x) + \cos(x)) i
9483928192 \cos(2x)+i\sin(2x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
9485384858 \nabla^2 E(\vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E(\vec{r})\exp(i \omega t)
9485747245 \sqrt{\frac{2}{W}} = a
9485747246 -\sqrt{\frac{2}{W}} = a
9492920340 y = \cos(x)+i \sin(x)
9494829190 k  
9495857278 \psi(x=W) = 0
9499428242 E(\vec{r},t) = E(\vec{r})\exp(i \omega t)
9499959299 a + k = b + k  
9582958293 x = \sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
9582958294 x+(b/(2a)) = \sqrt{(b/(2a))^2 - (c/a)}
9584821911 c + d = a  
9585727710 \psi(x=0) = 0
9596004948 x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
9848292229 dy = y i dx
9848294829 \frac{d}{dx} y = y i
9858028950 \frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
9889984281 2(\sin(x))^2 = 1-\cos(2x)
9919999981 \rho = 0
9958485859 \frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
9988949211 (\sin(x))^2 = \frac{1-\cos(2x)}{2}
9991999979 \vec{\nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
9999998870 \frac{\vec{p}}{\hbar} = \vec{k}
9999999870 \frac{p}{\hbar} = k
9999999950 \beta = 1/(k_{Boltzmann}T)  
9999999951 \langle x | k \rangle = \frac{\exp(i k x)}{\sqrt{2\pi}}  
9999999952 f(x) \delta(x-a) = f(a)
 
9999999953 \int_{-\infty}^{\infty} \delta(x) dx = 1
 
9999999954 c = 1/\sqrt{\epsilon_0 \mu_0}  
9999999955 \vec{E} = -\vec{\nabla} \Psi
 
9999999956 \vec{F} = \frac{\partial}{\partial t}\vec{p}  
9999999957 \vec{F} = -\vec{\nabla} V
 
9999999960 \hbar = h/(2 \pi)
9999999961 \frac{E}{\hbar} = \omega
9999999962 p = \hbar k
9999999963 \lambda = h/p
 
9999999964 \omega = c k  
9999999965 E = \omega \hbar
9999999966 \vec{L} = \vec{r}\times\vec{p}
 
9999999967 \vec{S} = \frac{1}{\mu_0} \vec{E}\times \vec{B}
 
9999999968 x = \frac{-b-\sqrt{b^2-4ac}}{2a}
9999999969 x = \frac{-b+\sqrt{b^2-4ac}}{2a}
9999999970 \eta_1 \sin\theta_1 = \eta_2 \sin\theta_2
 
9999999971 {\cal H} = \frac{p^2}{2m}+V
 
9999999972 {\cal H} |\psi\rangle = E |\psi\rangle
 
9999999973 \left( \Delta A \right)^2 = \langle A^2 \rangle - \langle A \rangle^2
 
9999999974 \langle \psi| \hat{A} |\psi\rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx
 
9999999975 \langle \psi| \hat{A} |\psi\rangle = \langle a \rangle
9999999976 \hat{p} = -i \hbar \frac{\partial }{\partial x}  
9999999977 [\hat{x},\hat{p}] = i \hbar  
9999999978 \vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}
 
9999999979 \vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}  
9999999980 \vec{\nabla} \cdot \vec{B} = 0
 
9999999981 \vec{\nabla} \cdot \vec{E} = \rho/\epsilon_0
9999999982 V = I R + Q/C + L \frac{\partial I}{\partial t}  
9999999983 C = V A/d
 
9999999984 Q = C V
 
9999999985 V = I R
 
9999999986 \left[cos⁡θsin⁡θ−sin⁡θcos⁡θ\right]\left[AxAy\right] = \left[Ax′Ay′\right]
 
9999999987 1 atmosphere = 101.325 Pascal
 
9999999988 1 atmosphere = 14.7 pounds/(inch^2)
 
9999999989 mass_{electron} = 511000 electronVolts/(q^2)
 
9999999990 Tesla = 10000*Gauss
 
9999999991 Pascal = Newton / (meter^2)
 
9999999992 Tesla = Newton / (Ampere*meter)
 
9999999993 Farad = Coulumb / Volt
 
9999999995 Volt = Joule / Coulumb
 
9999999996 Coulomb = Ampere / second
 
9999999997 Watt = Joule / second
 
9999999998 Joule = Newton*meter
 
9999999999 Newton = kilogram*meter/(second^2)
 
4961662865 x
9110536742 2 x
8483686863 \sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x)\right)
3470587782 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
8642992037 2
9894826550 2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \left(\exp(i x)+\exp(-i x)\right)
4257214632 \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)  
8699789241 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
9180861128 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
2405307372 \sin(2 x) = 2 \sin(x) \cos(x)
3268645065 x
9350663581 \pi
8332931442 \exp(i \pi) = \cos(\pi)+i \sin(\pi)
6885625907 \exp(i \pi) = -1 + i 0
3331824625 \exp(i \pi) = -1
4901237716 1
2501591100 \exp(i \pi) + 1 = 0
5136652623 E = KE + PE mechanical energy is the sum of the potential plus kinetic energies
7915321076 E, KE, PE  
3192374582 E_2, KE_2, PE_2
 
7875206161 E_2 = KE_2 + PE_2
4229092186 E, KE, PE  
1490821948 E_1, KE_1, PE_1
 
4303372136 E_1 = KE_1 + PE_1
5514556106 E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
8357234146 KE = \frac{1}{2} m v^2 kinetic energy Kinetic_energy
3658066792 KE_2, v_2
 
8082557839 KE, v  
7676652285 KE_2 = \frac{1}{2} m v_2^2
2790304597 KE_1, v_1
 
9880327189 KE, v  
4928007622 KE_1 = \frac{1}{2} m v_1^2
5733146966 KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
6554292307 t
4270680309 \frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
5781981178 x^2 - y^2 = (x+y)(x-y) difference of squares Difference_of_two_squares
5946759357 v_2, v_1
 
6675701064 x, y  
4648451961 v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
9356924046 \frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
9397152918 v = \frac{v_1 + v_2}{2} average velocity
2857430695 a = \frac{v_2 - v_1}{t}
acceleration
7735737409 \frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
4784793837 \frac{KE_2 - KE_1}{t} = m v a
5345738321 F = m a Newton's second law of motion Newton%27s_laws_of_motion#Newton's_second_law
2186083170 \frac{KE_2 - KE_1}{t} = v F
6715248283 PE = -F x potential energy Potential_energy
3852078149 PE_2, x_2
 
7388921188 PE, x  
2431507955 PE_2 = -F x_2
3412641898 PE_1, x_1
 
9937169749 PE, x  
4669290568 PE_1 = -F x_1
7734996511 PE_2 - PE_1 = -F ( x_2 - x_1 )
2016063530 t
7267155233 \frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
9337785146 v = \frac{x_2 - x_1}{t} average velocity
4872970974 \frac{PE_2 - PE_1}{t} = -F v
2081689540 t
2770069250 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
3591237106 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
1772416655 \frac{E_2 - E_1}{t} = v F - F v
1809909100 \frac{E_2 - E_1}{t} = 0
5778176146 t
3806977900 E_2 - E_1 = 0
5960438249 E_1
8558338742 E_2 = E_1
8945218208 \theta_{Brewster} + \theta_{refracted} = 90^{\circ} based on figure 34-27 on page 824 in 2001_HRW
9025853427 \theta_{Brewster}
1310571337 \theta_{refracted} = 90^{\circ} - \theta_{Brewster}
6450985774 n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 ) Law of Refraction eq 34-44 on page 819 in 2001_HRW
1779497048 \theta_{Brewster}, \theta_{refraction}  
7322344455 \theta_1, \theta_2
 
2575937347 n_1 \sin( \theta_{Brewster} ) = n_2 \sin( \theta_{refraction} )
7696214507 n_1 \sin( \theta_{Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{Brewster} )
8588429722 \sin( 90^{\circ} - x ) = \cos( x )
7375348852 \theta_{Brewster}
1512581563 x
6831637424 \sin( 90^{\circ} - \theta_{Brewster} ) = \cos( \theta_{Brewster} )
3061811650 n_1 \sin( \theta_{Brewster} ) = n_2 \cos( \theta_{Brewster} )
7857757625 n_1
9756089533 \sin( \theta_{Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{Brewster} )
5632428182 \cos( \theta_{Brewster} )
2768857871 \frac{\sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )} = \frac{n_2}{n_1}
4968680693 \tan( x ) = \frac{ \sin( x )}{\cos( x )}
7321695558 \theta_{Brewster}
9906920183 x
4501377629 \tan( \theta_{Brewster} ) = \frac{ \sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )}
3417126140 \tan( \theta_{Brewster} ) = \frac{ n_2 }{ n_1 }
5453995431 \arctan{ x }
6023986360 x
8495187962 \theta_{Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
9040079362 f
9565166889 T
5426308937 v = \frac{d}{t}
7476820482 C
1277713901 d
6946088325 v = \frac{C}{t}
6785303857 C = 2 \pi r
4816195267 C, r
 
2113447367 C_{\rm{Earth\ orbit}}, r_{\rm{Earth\ orbit}}
 
6348260313 C_{\rm{Earth\ orbit}} = 2 \pi r_{\rm{Earth\ orbit}}
3847777611 C, v, t  
6406487188 C_{\rm{Earth\ orbit}}, v_{\rm{Earth\ orbit}}, t_{\rm{Earth\ orbit}}
 
3046191961 v_{\rm{Earth\ orbit}} = \frac{C_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
3080027960 v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
7175416299 t_{\rm{Earth\ orbit}} = 1 {\rm{year}}
3219318145 \frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}
8721295221 t_{\rm{Earth\ orbit}} = 3.16 10^7 {\rm{seconds}}
4593428198 v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{3.16\ 10^7 {\rm seconds}}
3472836147 r_{\rm{Earth\ orbit}} = 1.496\ 10^8 {\rm km}
6998364753 v_{\rm{Earth\ orbit}} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
4180845508 v_{\rm{Earth\ orbit}} = 29.8 \frac{{\rm km}}{{\rm sec}}
4662369843 x' = \gamma (x - v t)
2983053062 x = \gamma (x' + v t')
3426941928 x = \gamma ( \gamma (x - v t) + v t' )
2096918413 x = \gamma ( \gamma x - \gamma v t + v t' )
7741202861 x = \gamma^2 x - \gamma^2 v t + \gamma v t'
7337056406 \gamma^2 x
4139999399 x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
3495403335 x
9031609275 x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
8014566709 \gamma^2 v t
9409776983 x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
2226340358 \gamma v
6417705759 \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = \gamma v t'
 
8730201316 \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'
first term was multiplied by \gamma/\gamma
1974334644 \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
5148266645 t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
4287102261 x^2 + y^2 + z^2 = c^2 t^2 describes a spherical wavefront
1201689765 x'^2 + y'^2 + z'^2 = c^2 t'^2
describes a spherical wavefront for an observer in a moving frame of reference
7057864873 y' = y
frame of reference is moving only along x direction
8515803375 z' = z
frame of reference is moving only along x direction
6153463771 3
 
4746576736 asdf
 
1027270623 minga
 
6101803533 ming
 
9231495607 a = b
 
8664264194 b + 2
 
9805063945 \gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2
4057686137 C \to C_{\rm{Earth\ orbit}}
2346150725 r \to r_{\rm{Earth\ orbit}}
9753878784 v \to v_{\rm{Earth\ orbit}}
8135396036 t \to t_{\rm{Earth\ orbit}}
2773628333 \theta_1 \to \theta_{Brewster}
7154592211 \theta_2 \to \theta_{\rm refracted}
3749492596 E \to E_1
1258245373 E \to E_2
4147101187 KE \to KE_1
3809726424 PE \to PE_1
6383056612 KE \to KE_2
5075406409 PE \to PE_2
5398681503 v \to v_1
9305761407 v \to v_2
4218009993 x \to x_1
4188639044 x \to x_2
8173074178 x \to v_2
1025759423 y \to v_1
1935543849 \gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2
1586866563 \left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)
3182633789 \gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
1916173354 -\gamma^2 v^2 + c^2 \gamma^2 = c^2
2076171250 -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0
5284610349 \gamma^2
2417941373 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
5787469164 1 - \gamma^2
1639827492 - c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
5669500954 v^2 \gamma^2
5763749235 -c^2 + c^2 \gamma^2 = v^2 \gamma^2
6408214498 c^2
2999795755 c^2 \gamma^2 = v^2 \gamma^2 + c^2
3412946408 v^2 \gamma^2
2542420160 c^2 \gamma^2 - v^2 \gamma^2 = c^2
7743841045 \gamma^2
7513513483 \gamma^2 (c^2 - v^2) = c^2
8571466509 c^2 - \gamma^2
7906112355 \gamma^2 = \frac{c^2}{c^2 - \gamma^2}
3060139639 1/2
 
1528310784 \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
8360117126 \gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}} not a physically valid result in this context
3366703541 a = \frac{v - v_0}{t} acceleration is the average change in speed over a duration
7083390553 t
4748157455 a t = v - v_0
6417359412 v_0
4798787814 a t + v_0 = v
3462972452 v = v_0 + a t
3411994811 v_{\rm ave} = \frac{d}{t}
6175547907 v_{\rm ave} = \frac{v + v_0}{2}
9897284307 \frac{d}{t} = \frac{v + v_0}{2}
8865085668 t
8706092970 d = \left(\frac{v + v_0}{2}\right)t
7011114072 d = \frac{(v_0 + a t) + v_0}{2} t
1265150401 d = \frac{2 v_0 + a t}{2} t
9658195023 d = v_0 t + \frac{1}{2} a t^2
5799753649 2
7215099603 v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
5144263777 v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
7939765107 v^2 = v_0^2 + 2 a d
6729698807 v_0
9759901995 v - v_0 = a t
2242144313 a
1967582749 t = \frac{v - v_0}{a}
5733721198 d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
5611024898 d = \frac{1}{2 a} (v^2 - v_0^2)
5542390646 2 a
8269198922 2 a d = v^2 - v_0^2
9070454719 v_0^2
4948763856 2 a d + v_0^2 = v^2
9645178657 a t
6457044853 v - a t = v_0
1259826355 d = (v - a t) t + \frac{1}{2} a t^2
4580545876 d = v t - a t^2 + \frac{1}{2} a t^2
6421241247 d = v t - \frac{1}{2} a t^2
4182362050 Z = |Z| \exp( i \theta ) Z \in \mathbb{C}
1928085940 Z^* = |Z| \exp( -i \theta )
9191880568 Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
3350830826 Z Z^* = |Z|^2
8396997949 I = | A + B |^2 intensity of two waves traveling opposite directions on same path
4437214608 Z
5623794884 A + B
2236639474 (A + B)(A + B)^* = |A + B|^2
1020854560 I = (A + B)(A + B)^*
6306552185 I = (A + B)(A^* + B^*)
8065128065 I = A A^* + B B^* + A B^* + B A^*
9761485403 Z
8710504862 A
4075539836 A A^* = |A|^2
6529120965 B
1511199318 Z
7107090465 B B^* = |B|^2
5125940051 I = |A|^2 + B B^* + A B^* + B A^*
1525861537 I = |A|^2 + |B|^2 + A B^* + B A^*
1742775076 Z \to B
7607271250 \theta \to \phi
4192519596 B = |B| \exp(i \phi)
2064205392 A
1894894315 Z
1357848476 A = |A| \exp(i \theta)
6555185548 A^* = |A| \exp(-i \theta)
4504256452 B^* = |B| \exp(-i \phi)
7621705408 I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
3085575328 I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
4935235303 x
2293352649 \theta - \phi
3660957533 \cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
3967985562 2
2700934933 2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
8497631728 I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
6774684564 \theta = \phi for coherent waves
2099546947 I = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )  
2719691582 |A| = |B| in a loop
3257276368 I = |A|^2 + |A|^2 + |A| |A| 2
 
8283354808 I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
8046208134 I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
1172039918 I_{\rm coherent} = 4 |A|^2
8602221482 \langle \cos(\theta - \phi) \rangle = 0 incoherent light source
6240206408 I_{\rm incoherent} = |A|^2 + |B|^2
6529793063 I_{\rm incoherent} = |A|^2 + |A|^2
3060393541 I_{\rm incoherent} = 2|A|^2
6556875579 \frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
8750500372 f
 
7543102525 g
 
8267133829 a = b
 
7968822469 c = d
 
3169580383 \vec{a} = \frac{d\vec{v}}{dt} acceleration is the change in speed over a duration
8602512487 \vec{a} = a_x \hat{x} + a_y \hat{y} decompose acceleration into two components
4158986868 a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
5349866551 \vec{v} = v_x \hat{x} + v_y \hat{y}
7729413831 a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
1819663717 a_x = \frac{d}{dt} v_x
8228733125 a_y = \frac{d}{dt} v_y
9707028061 a_x = 0
2741489181 a_y = -g
3270039798 2
8880467139 2
8750379055 0 = \frac{d}{dt} v_x
1977955751 -g = \frac{d}{dt} v_y
6672141531 dt
1702349646 -g dt = d v_y
8584698994 -g \int dt = \int d v_y
9973952056 -g t = v_y - v_{0, y}
4167526462 v_{0, y}
6572039835 -g t + v_{0, y} = v_y
7252338326 v_y = \frac{dy}{dt}
6204539227 -g t + v_{0, y} = \frac{dy}{dt}
1614343171 dt
8145337879 -g t dt + v_{0, y} dt = dy
8808860551 -g \int t dt + v_{0, y} \int dt = \int dy
2858549874 - \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
6098638221 y_0
2461349007 - \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
8717193282 dt
1166310428 0 dt = d v_x
2366691988 \int 0 dt = \int d v_x
1676472948 0 = v_x - v_{0, x}
1439089569 v_{0, x}
6134836751 v_{0, x} = v_x
8460820419 v_x = \frac{dx}{dt}
7455581657 v_{0, x} = \frac{dx}{dt}
8607458157 dt
1963253044 v_{0, x} dt = dx
3676159007 v_{0, x} \int dt = \int dx
9882526611 v_{0, x} t = x - x_0
3182907803 x_0
8486706976 v_{0, x} t + x_0 = x
1306360899 x = v_{0, x} t + x_0
7049769409 2
9341391925 \vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
6410818363 \theta
7391837535 \cos(\theta) = \frac{v_{0, x}}{v_0}
7376526845 \sin(\theta) = \frac{v_{0, y}}{v_0}
5868731041 v_0
6083821265 v_0 \cos(\theta) = v_{0, x}
5438722682 x = v_0 t \cos(\theta) + x_0
5620558729 v_0
8949329361 v_0 \sin(\theta) = v_{0, y}
1405465835 y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
9862900242 y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
6050070428 v_{0, x}
3274926090 t = \frac{x - x_0}{v_{0, x}}
7354529102 y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0
8406170337 y \to y_f
2403773761 t \to t_f
5379546684 y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
5373931751 t = t_f
9112191201 y_f = 0
8198310977 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
1650441634 y_0 = 0
define coordinate system such that initial height is at origin
1087417579 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
4829590294 t_f
2086924031 0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
6974054946 \frac{1}{2} g t_f
1191796961 \frac{1}{2} g t_f = v_0 \sin(\theta)
2510804451 2/g
4778077984 t_f = \frac{2 v_0 \sin(\theta)}{g}
3273630811 x \to x_f
6732786762 t \to t_f
3485125659 x_f = v_0 t_f \cos(\theta) + x_0
4370074654 t = t_f
2378095808 x_f = x_0 + d
4268085801 x_0 + d = v_0 t_f \cos(\theta) + x_0
8072682558 x_0
7233558441 d = v_0 t_f \cos(\theta)
2297105551 d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
7587034465 \theta
7214442790 x
2519058903 \sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
8922441655 d = \frac{v_0^2}{g} \sin(2 \theta)
5667870149 \theta
1541916015 \theta = \frac{\pi}{4}
3607070319 d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
5353282496 d = \frac{v_0^2}{g}
0000040490 a^2
0000999900 b/(2a)
0001030901 \cos(x)
0001111111 (\sin(x))^2
0001209482 2 \pi
0001304952 \hbar
0001334112 W
0001921933 2 i
0002239424 2
0002338514 \vec{p}_{2}
0002342425 m/m
0002367209 1/2
 
0002393922 x
0002424922 a
0002436656 i \hbar
0002449291 b/(2a)
0002838490 b/(2a)
0002919191 \sin(-x)
0002929944 1/2
0002940021 2 \pi
0003232242 t
0003413423 \cos(-x)
0003747849 -1
0003838111 2
0003919391 x
0003949052 -x
0003949921 \hbar
0003954314 dx
0003981813 -\sin(x)
0004089571 2x
0004264724 y
0004307451 (b/(2a))^2
0004582412 x
0004829194 2
0004831494 a
0004849392 x
0004858592 h
0004934845 x
0004948585 a
0005395034 a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
0005626421 t
0005749291 f
0005938585 \frac{-\hbar^2}{2m}
0006466214 (\sin(x))^2
0006544644 t
0006563727 x
0006644853 c/a
0006656532 e
0007471778 2(\sin(x))^2
0007563791 i
0007636749 x
0007894942 (\sin(x))^2
0008837284 T
0008842811 \cos(2x)
0009458842 \psi(x)
0009484724 \frac{n \pi}{W}x
0009485857 a^2\frac{2}{W}
0009485858 \frac{2n\pi}{W}
0009492929 v du
0009587738 \psi
0009594995 1/2
 
0009877781 y
0203024440 1 = \int_0^W a \sin(\frac{n \pi}{W} x) \psi(x)^* dx
0404050504 \lambda = \frac{v}{f}
0439492440 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\right|_0^W
0934990943 k = \frac{2 \pi}{v T}
0948572140 \int \cos(a x) dx = \frac{1}{a}\sin(a x)
1010393913 \langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*
1010393944 x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
1010923823 k W = n \pi
1020010291 0 = a \sin(k W)
1020394900 p = h/\lambda
1020394902 E = h f
1029039903 p = m v
1029039904 p^2 = m^2 v^2
1158485859 \frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
1202310110 \frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1202312210 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1203938249 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
1248277773 \cos(2x) = 1-2(\sin(x))^2
1293913110 0 = b
1293923844 \lambda = v T
1314464131 \vec{\nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
1314864131 \vec{\nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
1395858355 x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
1492811142 f = x - d  
1492842000 \nabla \vec{x} = f(y)  
1636453295 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = - \nabla^2 \vec{E}
1638282134 \vec{p}_{before} = \vec{p}_{after}
1648958381 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{\nabla}\psi(\vec{r},t)\right)
1857710291 0 = a \sin(n \pi)
1858578388 \nabla^2 E(\vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E(\vec{r})\exp(i \omega t)
1858772113 k = \frac{n \pi}{W}
1934748140 \int |\psi(x)|^2 dx = 1
2029293929 \nabla^2 E(\vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r})\exp(i \omega t)
2103023049 \sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right)
2113211456 f = 1/T
2123139121 -\exp(-i x) = -\cos(x)+i \sin(x)
2131616531 T f = 1
2258485859 {\cal H} \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
2394240499 x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
2394853829 \exp(-i x) = \cos(-x)+i \sin(-x)
2394935831 ( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
2394935835 \left(\langle\psi| \hat{A} |\psi\rangle \right)^+ = \left(\langle a \rangle\right)^+
2395958385 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{-p^2}{\hbar} \psi(\vec{r},t)
2404934990 \langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
2494533900 \langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
2569154141 \vec{\nabla} \times \frac{\partial}{\partial t}\vec{H} = \epsilon_0 \frac{\partial^2 }{\partial t^2}\vec{E}  
2648958382 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p}\psi(\vec{r},t)\right)
2848934890 \langle a \rangle^* = \langle a \rangle
2944838499 \psi(x) = a \sin(\frac{n \pi}{W} x)
3121234211 \frac{k}{2\pi} = \lambda
3121234212 p = \frac{h k}{2\pi}
3121513111 k = \frac{2 \pi}{\lambda}
3131111133 T = 1 / f
3131211131 \omega = 2 \pi f
3132131132 \omega = \frac{2\pi}{T}
3147472131 \frac{\omega}{2 \pi} = f
3285732911 (\cos(x))^2 = 1-(\sin(x))^2
3485475729 \nabla^2 E(\vec{r}) = - \frac{\omega^2}{c^2} E(\vec{r})
3585845894 \langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
3829492824 \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right) = \cos(x)
3924948349 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
3942849294 \exp(i x)-\exp(-i x) = 2 i \sin(x)
3943939590 x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
3947269979 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
3948571256 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi(\vec{r},t)
3948572230 \vec{\nabla}\psi(\vec{r},t) = \psi_0 \vec{\nabla}\exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)  
3948574224 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\vec{k}\cdot\vec{r} - \omega t \right) \right)
3948574226 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
3948574228 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3948574230 \psi(\vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
3948574233 \frac{\partial}{\partial t}\psi(\vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3948574235 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)  
3951205425 \vec{p}_{after} = \vec{p}_{1}
4147472132 E = \frac{h \omega}{2 \pi}
4298359835 E = \frac{1}{2}m v^2
4298359845 E = \frac{1}{2m}m^2 v^2
4298359851 E = \frac{p^2}{2m}
4341171256 i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{p^2}{2 m} \psi(\vec{r},t)
4348571256 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi(\vec{r},t)
4394958389 \vec{\nabla}\cdot \left(\vec{\nabla}\psi(\vec{r},t)\right) = \frac{i}{\hbar} \vec{\nabla}\cdot\left( \vec{p} \psi(\vec{r},t)\right)
4585828572 \epsilon_0 \mu_0 = \frac{1}{c^2}
4585932229 \cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
4598294821 \exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
4638429483 \exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
4742644828 \exp(i x)+\exp(-i x) = 2 \cos(x)
4827492911 \cos(2x)+(\sin(x))^2 = 1-(\sin(x))^2
4838429483 \exp(2 i x) = \cos(2 x)+i \sin(2 x)
4843995999 \frac{1}{2 i}\left(\exp(i x)-\exp(-i x)\right) = \sin(x)
4857472413 1 = \int \psi(x)\psi(x)^* dx
4857475848 \frac{1}{a^2} = \frac{W}{2}
4858429483 \exp(i x)\exp(i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))  
4923339482 i x = \log(y)
4928239482 \log(y) = i x
4928923942 a = b  
4929218492 a+b = c  
4929482992 b = 2
 
4938429482 \exp(-i x) = \cos(x)+i \sin(-x)
4938429483 \exp(i x) = \cos(x)+i \sin(x)
4938429484 \exp(-i x) = \cos(x)-i \sin(x)
4943571230 \vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
4948934890 \langle \psi| \hat{A} |\psi\rangle = \langle a \rangle^*
4949359835 \langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
4954839242 \cos(2x)+i\sin(2x) = (\cos(x)+i \sin(x))(\cos(x)+i \sin(x))  
4978429483 \exp(-i x) = \cos(-x)+i \sin(-x)  
4984892984 a+2 = c
 
4985825552 \nabla^2 E(\vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E(\vec{r})\exp(i \omega t)
5530148480 \vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
5727578862 \frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
5832984291 (\sin(x))^2 + (\cos(x))^2 = 1
5857434758 \int a dx = a x
5868688585 \frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = \frac{p^2}{2m} \psi(\vec{r},t)
5900595848 k = \frac{\omega}{v}
5928285821 x^2 + 2x(b/(2a)) + (b/(2a))^2 = (x+(b/(2a)))^2
5928292841 x^2 + (b/a)x + (b/(2a))^2 = -c/a + (b/(2a))^2
5938459282 x^2 + (b/a)x = -c/a
5958392859 x^2 + (b/a)x+(c/a) = 0
5959282914 x^2 + x(b/a) + (b/(2a))^2 = (x+(b/(2a)))^2
5982958248 x = -\sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
5982958249 x+(b/(2a)) = -\sqrt{(b/(2a))^2 - (c/a)}
5985371230 \vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi(\vec{r},t)
6742123016 \vec{p}_{electron}\cdot\vec{p}_{electron} = (\vec{p}_{1}\cdot\vec{p}_{1})+(\vec{p}_{2}\cdot\vec{p}_{2})-2(\vec{p}_{1}\cdot\vec{p}_{2})
7466829492 \vec{\nabla} \cdot \vec{E} = 0
7564894985 \int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
7572664728 \cos(2x)+2(\sin(x))^2 = 1
7575738420 \left(\sin\left(\frac{n \pi}{W}x\right)\right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
7575859295 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859300 \epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{\nabla} \times \vec{E} )_k = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859302 \epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859304 \epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
7575859306 \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859308 \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859310 \hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859312 \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7917051060 \vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
8139187332 \vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
8257621077 \vec{p}_{before} = \vec{p}_{1}
8311458118 \vec{p}_{after} = \vec{p}_{2}+\vec{p}_{electron}
8399484849 \langle x^2 -2x\langle x \rangle+\langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
8484544728 -a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
8485757728 a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
8485867742 \frac{2}{W} = a^2
8489593958 d(u v) = u dv + v du
8489593960 d(u v) - v du = u dv
8489593962 u dv = d(u v) - v du
8489593964 \int u dv = u v - \int v du
8494839423 \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
8572657110 1 = \int |\psi(x)|^2 dx
8572852424 \vec{E} = E(\vec{r},t)
8575746378 \int \frac{1}{2} dx = \frac{1}{2} x
8575748999 \frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x)\right) = -k^2 \left(a \sin(kx) + b \cos(kx)\right)
8576785890 1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
8577275751 0 = a \sin(0) + b\cos(0)
8582885111 \psi(x) = a \sin(kx) + b \cos(kx)
8582954722 x^2 + 2xh + h^2 = (x+h)^2
8849289982 \psi(x)^* = a \sin(\frac{n \pi}{W} x)
8889444440 1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right)\right)^2 dx
9059289981 \psi(x) = a \sin(k x)
9285928292 ax^2 + bx + c = 0
9291999979 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t}
9294858532 \hat{A}^+ = \hat{A}
9385938295 (x+(b/(2a)))^2 = -(c/a) + (b/(2a))^2
9393939991 \psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9393939992 \psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9394939493 \nabla^2 E(\vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r},t)
9429829482 \frac{d}{dx} y = -\sin(x) + i\cos(x)
9482113948 \frac{dy}{y} = i dx
9482438243 (\cos(x))^2 = \cos(2x)+(\sin(x))^2
9482923849 \exp(i x) = y
9482928242 \cos(2x) = (\cos(x))^2 - (\sin(x))^2
9482928243 \cos(2x)+(\sin(x))^2 = (\cos(x))^2
9482943948 \log(y) = i dx
9482948292 b = c
 
9482984922 \frac{d}{dx} y = (i\sin(x) + \cos(x)) i
9483928192 \cos(2x)+i\sin(2x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
9485384858 \nabla^2 E(\vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E(\vec{r})\exp(i \omega t)
9485747245 \sqrt{\frac{2}{W}} = a
9485747246 -\sqrt{\frac{2}{W}} = a
9492920340 y = \cos(x)+i \sin(x)
9494829190 k  
9495857278 \psi(x=W) = 0
9499428242 E(\vec{r},t) = E(\vec{r})\exp(i \omega t)
9499959299 a + k = b + k  
9582958293 x = \sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
9582958294 x+(b/(2a)) = \sqrt{(b/(2a))^2 - (c/a)}
9584821911 c + d = a  
9585727710 \psi(x=0) = 0
9596004948 x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
9848292229 dy = y i dx
9848294829 \frac{d}{dx} y = y i
9858028950 \frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
9889984281 2(\sin(x))^2 = 1-\cos(2x)
9919999981 \rho = 0
9958485859 \frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
9988949211 (\sin(x))^2 = \frac{1-\cos(2x)}{2}
9991999979 \vec{\nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
9999998870 \frac{\vec{p}}{\hbar} = \vec{k}
9999999870 \frac{p}{\hbar} = k
9999999950 \beta = 1/(k_{Boltzmann}T)  
9999999951 \langle x | k \rangle = \frac{\exp(i k x)}{\sqrt{2\pi}}  
9999999952 f(x) \delta(x-a) = f(a)
 
9999999953 \int_{-\infty}^{\infty} \delta(x) dx = 1
 
9999999954 c = 1/\sqrt{\epsilon_0 \mu_0}  
9999999955 \vec{E} = -\vec{\nabla} \Psi
 
9999999956 \vec{F} = \frac{\partial}{\partial t}\vec{p}  
9999999957 \vec{F} = -\vec{\nabla} V
 
9999999960 \hbar = h/(2 \pi)
9999999961 \frac{E}{\hbar} = \omega
9999999962 p = \hbar k
9999999963 \lambda = h/p
 
9999999964 \omega = c k  
9999999965 E = \omega \hbar
9999999966 \vec{L} = \vec{r}\times\vec{p}
 
9999999967 \vec{S} = \frac{1}{\mu_0} \vec{E}\times \vec{B}
 
9999999968 x = \frac{-b-\sqrt{b^2-4ac}}{2a}
9999999969 x = \frac{-b+\sqrt{b^2-4ac}}{2a}
9999999970 \eta_1 \sin\theta_1 = \eta_2 \sin\theta_2
 
9999999971 {\cal H} = \frac{p^2}{2m}+V
 
9999999972 {\cal H} |\psi\rangle = E |\psi\rangle
 
9999999973 \left( \Delta A \right)^2 = \langle A^2 \rangle - \langle A \rangle^2
 
9999999974 \langle \psi| \hat{A} |\psi\rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx
 
9999999975 \langle \psi| \hat{A} |\psi\rangle = \langle a \rangle
9999999976 \hat{p} = -i \hbar \frac{\partial }{\partial x}  
9999999977 [\hat{x},\hat{p}] = i \hbar  
9999999978 \vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}
 
9999999979 \vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}  
9999999980 \vec{\nabla} \cdot \vec{B} = 0
 
9999999981 \vec{\nabla} \cdot \vec{E} = \rho/\epsilon_0
9999999982 V = I R + Q/C + L \frac{\partial I}{\partial t}  
9999999983 C = V A/d
 
9999999984 Q = C V
 
9999999985 V = I R
 
9999999986 \left[cos⁡θsin⁡θ−sin⁡θcos⁡θ\right]\left[AxAy\right] = \left[Ax′Ay′\right]
 
9999999987 1 atmosphere = 101.325 Pascal
 
9999999988 1 atmosphere = 14.7 pounds/(inch^2)
 
9999999989 mass_{electron} = 511000 electronVolts/(q^2)
 
9999999990 Tesla = 10000*Gauss
 
9999999991 Pascal = Newton / (meter^2)
 
9999999992 Tesla = Newton / (Ampere*meter)
 
9999999993 Farad = Coulumb / Volt
 
9999999995 Volt = Joule / Coulumb
 
9999999996 Coulomb = Ampere / second
 
9999999997 Watt = Joule / second
 
9999999998 Joule = Newton*meter
 
9999999999 Newton = kilogram*meter/(second^2)
 
4961662865 x
9110536742 2 x
8483686863 \sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x)\right)
3470587782 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
8642992037 2
9894826550 2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \left(\exp(i x)+\exp(-i x)\right)
4257214632 \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)  
8699789241 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
9180861128 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
2405307372 \sin(2 x) = 2 \sin(x) \cos(x)
3268645065 x
9350663581 \pi
8332931442 \exp(i \pi) = \cos(\pi)+i \sin(\pi)
6885625907 \exp(i \pi) = -1 + i 0
3331824625 \exp(i \pi) = -1
4901237716 1
2501591100 \exp(i \pi) + 1 = 0
5136652623 E = KE + PE mechanical energy is the sum of the potential plus kinetic energies
7915321076 E, KE, PE  
3192374582 E_2, KE_2, PE_2
 
7875206161 E_2 = KE_2 + PE_2
4229092186 E, KE, PE  
1490821948 E_1, KE_1, PE_1
 
4303372136 E_1 = KE_1 + PE_1
5514556106 E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
8357234146 KE = \frac{1}{2} m v^2 kinetic energy Kinetic_energy
3658066792 KE_2, v_2
 
8082557839 KE, v  
7676652285 KE_2 = \frac{1}{2} m v_2^2
2790304597 KE_1, v_1
 
9880327189 KE, v  
4928007622 KE_1 = \frac{1}{2} m v_1^2
5733146966 KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
6554292307 t
4270680309 \frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
5781981178 x^2 - y^2 = (x+y)(x-y) difference of squares Difference_of_two_squares
5946759357 v_2, v_1
 
6675701064 x, y  
4648451961 v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
9356924046 \frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
9397152918 v = \frac{v_1 + v_2}{2} average velocity
2857430695 a = \frac{v_2 - v_1}{t}
acceleration
7735737409 \frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
4784793837 \frac{KE_2 - KE_1}{t} = m v a
5345738321 F = m a Newton's second law of motion Newton%27s_laws_of_motion#Newton's_second_law
2186083170 \frac{KE_2 - KE_1}{t} = v F
6715248283 PE = -F x potential energy Potential_energy
3852078149 PE_2, x_2
 
7388921188 PE, x  
2431507955 PE_2 = -F x_2
3412641898 PE_1, x_1
 
9937169749 PE, x  
4669290568 PE_1 = -F x_1
7734996511 PE_2 - PE_1 = -F ( x_2 - x_1 )
2016063530 t
7267155233 \frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
9337785146 v = \frac{x_2 - x_1}{t} average velocity
4872970974 \frac{PE_2 - PE_1}{t} = -F v
2081689540 t
2770069250 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
3591237106 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
1772416655 \frac{E_2 - E_1}{t} = v F - F v
1809909100 \frac{E_2 - E_1}{t} = 0
5778176146 t
3806977900 E_2 - E_1 = 0
5960438249 E_1
8558338742 E_2 = E_1
8945218208 \theta_{Brewster} + \theta_{refracted} = 90^{\circ} based on figure 34-27 on page 824 in 2001_HRW
9025853427 \theta_{Brewster}
1310571337 \theta_{refracted} = 90^{\circ} - \theta_{Brewster}
6450985774 n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 ) Law of Refraction eq 34-44 on page 819 in 2001_HRW
1779497048 \theta_{Brewster}, \theta_{refraction}  
7322344455 \theta_1, \theta_2
 
2575937347 n_1 \sin( \theta_{Brewster} ) = n_2 \sin( \theta_{refraction} )
7696214507 n_1 \sin( \theta_{Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{Brewster} )
8588429722 \sin( 90^{\circ} - x ) = \cos( x )
7375348852 \theta_{Brewster}
1512581563 x
6831637424 \sin( 90^{\circ} - \theta_{Brewster} ) = \cos( \theta_{Brewster} )
3061811650 n_1 \sin( \theta_{Brewster} ) = n_2 \cos( \theta_{Brewster} )
7857757625 n_1
9756089533 \sin( \theta_{Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{Brewster} )
5632428182 \cos( \theta_{Brewster} )
2768857871 \frac{\sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )} = \frac{n_2}{n_1}
4968680693 \tan( x ) = \frac{ \sin( x )}{\cos( x )}
7321695558 \theta_{Brewster}
9906920183 x
4501377629 \tan( \theta_{Brewster} ) = \frac{ \sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )}
3417126140 \tan( \theta_{Brewster} ) = \frac{ n_2 }{ n_1 }
5453995431 \arctan{ x }
6023986360 x
8495187962 \theta_{Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
9040079362 f
9565166889 T
5426308937 v = \frac{d}{t}
7476820482 C
1277713901 d
6946088325 v = \frac{C}{t}
6785303857 C = 2 \pi r
4816195267 C, r
 
2113447367 C_{\rm{Earth\ orbit}}, r_{\rm{Earth\ orbit}}
 
6348260313 C_{\rm{Earth\ orbit}} = 2 \pi r_{\rm{Earth\ orbit}}
3847777611 C, v, t  
6406487188 C_{\rm{Earth\ orbit}}, v_{\rm{Earth\ orbit}}, t_{\rm{Earth\ orbit}}
 
3046191961 v_{\rm{Earth\ orbit}} = \frac{C_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
3080027960 v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
7175416299 t_{\rm{Earth\ orbit}} = 1 {\rm{year}}
3219318145 \frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}
8721295221 t_{\rm{Earth\ orbit}} = 3.16 10^7 {\rm{seconds}}
4593428198 v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{3.16\ 10^7 {\rm seconds}}
3472836147 r_{\rm{Earth\ orbit}} = 1.496\ 10^8 {\rm km}
6998364753 v_{\rm{Earth\ orbit}} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
4180845508 v_{\rm{Earth\ orbit}} = 29.8 \frac{{\rm km}}{{\rm sec}}
4662369843 x' = \gamma (x - v t)
2983053062 x = \gamma (x' + v t')
3426941928 x = \gamma ( \gamma (x - v t) + v t' )
2096918413 x = \gamma ( \gamma x - \gamma v t + v t' )
7741202861 x = \gamma^2 x - \gamma^2 v t + \gamma v t'
7337056406 \gamma^2 x
4139999399 x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
3495403335 x
9031609275 x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
8014566709 \gamma^2 v t
9409776983 x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
2226340358 \gamma v
6417705759 \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = \gamma v t'
 
8730201316 \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'
first term was multiplied by \gamma/\gamma
1974334644 \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
5148266645 t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
4287102261 x^2 + y^2 + z^2 = c^2 t^2 describes a spherical wavefront
1201689765 x'^2 + y'^2 + z'^2 = c^2 t'^2
describes a spherical wavefront for an observer in a moving frame of reference
7057864873 y' = y
frame of reference is moving only along x direction
8515803375 z' = z
frame of reference is moving only along x direction
6153463771 3
 
4746576736 asdf
 
1027270623 minga
 
6101803533 ming
 
9231495607 a = b
 
8664264194 b + 2
 
9805063945 \gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2
4057686137 C \to C_{\rm{Earth\ orbit}}
2346150725 r \to r_{\rm{Earth\ orbit}}
9753878784 v \to v_{\rm{Earth\ orbit}}
8135396036 t \to t_{\rm{Earth\ orbit}}
2773628333 \theta_1 \to \theta_{Brewster}
7154592211 \theta_2 \to \theta_{\rm refracted}
3749492596 E \to E_1
1258245373 E \to E_2
4147101187 KE \to KE_1
3809726424 PE \to PE_1
6383056612 KE \to KE_2
5075406409 PE \to PE_2
5398681503 v \to v_1
9305761407 v \to v_2
4218009993 x \to x_1
4188639044 x \to x_2
8173074178 x \to v_2
1025759423 y \to v_1
1935543849 \gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2
1586866563 \left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)
3182633789 \gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
1916173354 -\gamma^2 v^2 + c^2 \gamma^2 = c^2
2076171250 -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0
5284610349 \gamma^2
2417941373 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
5787469164 1 - \gamma^2
1639827492 - c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
5669500954 v^2 \gamma^2
5763749235 -c^2 + c^2 \gamma^2 = v^2 \gamma^2
6408214498 c^2
2999795755 c^2 \gamma^2 = v^2 \gamma^2 + c^2
3412946408 v^2 \gamma^2
2542420160 c^2 \gamma^2 - v^2 \gamma^2 = c^2
7743841045 \gamma^2
7513513483 \gamma^2 (c^2 - v^2) = c^2
8571466509 c^2 - \gamma^2
7906112355 \gamma^2 = \frac{c^2}{c^2 - \gamma^2}
3060139639 1/2
 
1528310784 \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
8360117126 \gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}} not a physically valid result in this context
3366703541 a = \frac{v - v_0}{t} acceleration is the average change in speed over a duration
7083390553 t
4748157455 a t = v - v_0
6417359412 v_0
4798787814 a t + v_0 = v
3462972452 v = v_0 + a t
3411994811 v_{\rm ave} = \frac{d}{t}
6175547907 v_{\rm ave} = \frac{v + v_0}{2}
9897284307 \frac{d}{t} = \frac{v + v_0}{2}
8865085668 t
8706092970 d = \left(\frac{v + v_0}{2}\right)t
7011114072 d = \frac{(v_0 + a t) + v_0}{2} t
1265150401 d = \frac{2 v_0 + a t}{2} t
9658195023 d = v_0 t + \frac{1}{2} a t^2
5799753649 2
7215099603 v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
5144263777 v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
7939765107 v^2 = v_0^2 + 2 a d
6729698807 v_0
9759901995 v - v_0 = a t
2242144313 a
1967582749 t = \frac{v - v_0}{a}
5733721198 d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
5611024898 d = \frac{1}{2 a} (v^2 - v_0^2)
5542390646 2 a
8269198922 2 a d = v^2 - v_0^2
9070454719 v_0^2
4948763856 2 a d + v_0^2 = v^2
9645178657 a t
6457044853 v - a t = v_0
1259826355 d = (v - a t) t + \frac{1}{2} a t^2
4580545876 d = v t - a t^2 + \frac{1}{2} a t^2
6421241247 d = v t - \frac{1}{2} a t^2
4182362050 Z = |Z| \exp( i \theta ) Z \in \mathbb{C}
1928085940 Z^* = |Z| \exp( -i \theta )
9191880568 Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
3350830826 Z Z^* = |Z|^2
8396997949 I = | A + B |^2 intensity of two waves traveling opposite directions on same path
4437214608 Z
5623794884 A + B
2236639474 (A + B)(A + B)^* = |A + B|^2
1020854560 I = (A + B)(A + B)^*
6306552185 I = (A + B)(A^* + B^*)
8065128065 I = A A^* + B B^* + A B^* + B A^*
9761485403 Z
8710504862 A
4075539836 A A^* = |A|^2
6529120965 B
1511199318 Z
7107090465 B B^* = |B|^2
5125940051 I = |A|^2 + B B^* + A B^* + B A^*
1525861537 I = |A|^2 + |B|^2 + A B^* + B A^*
1742775076 Z \to B
7607271250 \theta \to \phi
4192519596 B = |B| \exp(i \phi)
2064205392 A
1894894315 Z
1357848476 A = |A| \exp(i \theta)
6555185548 A^* = |A| \exp(-i \theta)
4504256452 B^* = |B| \exp(-i \phi)
7621705408 I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
3085575328 I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
4935235303 x
2293352649 \theta - \phi
3660957533 \cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
3967985562 2
2700934933 2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
8497631728 I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
6774684564 \theta = \phi for coherent waves
2099546947 I = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )  
2719691582 |A| = |B| in a loop
3257276368 I = |A|^2 + |A|^2 + |A| |A| 2
 
8283354808 I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
8046208134 I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
1172039918 I_{\rm coherent} = 4 |A|^2
8602221482 \langle \cos(\theta - \phi) \rangle = 0 incoherent light source
6240206408 I_{\rm incoherent} = |A|^2 + |B|^2
6529793063 I_{\rm incoherent} = |A|^2 + |A|^2
3060393541 I_{\rm incoherent} = 2|A|^2
6556875579 \frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
8750500372 f
 
7543102525 g
 
8267133829 a = b
 
7968822469 c = d
 
3169580383 \vec{a} = \frac{d\vec{v}}{dt} acceleration is the change in speed over a duration
8602512487 \vec{a} = a_x \hat{x} + a_y \hat{y} decompose acceleration into two components
4158986868 a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
5349866551 \vec{v} = v_x \hat{x} + v_y \hat{y}
7729413831 a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
1819663717 a_x = \frac{d}{dt} v_x
8228733125 a_y = \frac{d}{dt} v_y
9707028061 a_x = 0
2741489181 a_y = -g
3270039798 2
8880467139 2
8750379055 0 = \frac{d}{dt} v_x
1977955751 -g = \frac{d}{dt} v_y
6672141531 dt
1702349646 -g dt = d v_y
8584698994 -g \int dt = \int d v_y
9973952056 -g t = v_y - v_{0, y}
4167526462 v_{0, y}
6572039835 -g t + v_{0, y} = v_y
7252338326 v_y = \frac{dy}{dt}
6204539227 -g t + v_{0, y} = \frac{dy}{dt}
1614343171 dt
8145337879 -g t dt + v_{0, y} dt = dy
8808860551 -g \int t dt + v_{0, y} \int dt = \int dy
2858549874 - \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
6098638221 y_0
2461349007 - \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
8717193282 dt
1166310428 0 dt = d v_x
2366691988 \int 0 dt = \int d v_x
1676472948 0 = v_x - v_{0, x}
1439089569 v_{0, x}
6134836751 v_{0, x} = v_x
8460820419 v_x = \frac{dx}{dt}
7455581657 v_{0, x} = \frac{dx}{dt}
8607458157 dt
1963253044 v_{0, x} dt = dx
3676159007 v_{0, x} \int dt = \int dx
9882526611 v_{0, x} t = x - x_0
3182907803 x_0
8486706976 v_{0, x} t + x_0 = x
1306360899 x = v_{0, x} t + x_0
7049769409 2
9341391925 \vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
6410818363 \theta
7391837535 \cos(\theta) = \frac{v_{0, x}}{v_0}
7376526845 \sin(\theta) = \frac{v_{0, y}}{v_0}
5868731041 v_0
6083821265 v_0 \cos(\theta) = v_{0, x}
5438722682 x = v_0 t \cos(\theta) + x_0
5620558729 v_0
8949329361 v_0 \sin(\theta) = v_{0, y}
1405465835 y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
9862900242 y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
6050070428 v_{0, x}
3274926090 t = \frac{x - x_0}{v_{0, x}}
7354529102 y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0
8406170337 y \to y_f
2403773761 t \to t_f
5379546684 y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
5373931751 t = t_f
9112191201 y_f = 0
8198310977 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
1650441634 y_0 = 0
define coordinate system such that initial height is at origin
1087417579 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
4829590294 t_f
2086924031 0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
6974054946 \frac{1}{2} g t_f
1191796961 \frac{1}{2} g t_f = v_0 \sin(\theta)
2510804451 2/g
4778077984 t_f = \frac{2 v_0 \sin(\theta)}{g}
3273630811 x \to x_f
6732786762 t \to t_f
3485125659 x_f = v_0 t_f \cos(\theta) + x_0
4370074654 t = t_f
2378095808 x_f = x_0 + d
4268085801 x_0 + d = v_0 t_f \cos(\theta) + x_0
8072682558 x_0
7233558441 d = v_0 t_f \cos(\theta)
2297105551 d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
7587034465 \theta
7214442790 x
2519058903 \sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
8922441655 d = \frac{v_0^2}{g} \sin(2 \theta)
5667870149 \theta
1541916015 \theta = \frac{\pi}{4}
3607070319 d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
5353282496 d = \frac{v_0^2}{g}
0000040490 a^2
0000999900 b/(2a)
0001030901 \cos(x)
0001111111 (\sin(x))^2
0001209482 2 \pi
0001304952 \hbar
0001334112 W
0001921933 2 i
0002239424 2
0002338514 \vec{p}_{2}
0002342425 m/m
0002367209 1/2
 
0002393922 x
0002424922 a
0002436656 i \hbar
0002449291 b/(2a)
0002838490 b/(2a)
0002919191 \sin(-x)
0002929944 1/2
0002940021 2 \pi
0003232242 t
0003413423 \cos(-x)
0003747849 -1
0003838111 2
0003919391 x
0003949052 -x
0003949921 \hbar
0003954314 dx
0003981813 -\sin(x)
0004089571 2x
0004264724 y
0004307451 (b/(2a))^2
0004582412 x
0004829194 2
0004831494 a
0004849392 x
0004858592 h
0004934845 x
0004948585 a
0005395034 a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
0005626421 t
0005749291 f
0005938585 \frac{-\hbar^2}{2m}
0006466214 (\sin(x))^2
0006544644 t
0006563727 x
0006644853 c/a
0006656532 e
0007471778 2(\sin(x))^2
0007563791 i
0007636749 x
0007894942 (\sin(x))^2
0008837284 T
0008842811 \cos(2x)
0009458842 \psi(x)
0009484724 \frac{n \pi}{W}x
0009485857 a^2\frac{2}{W}
0009485858 \frac{2n\pi}{W}
0009492929 v du
0009587738 \psi
0009594995 1/2
 
0009877781 y
0203024440 1 = \int_0^W a \sin(\frac{n \pi}{W} x) \psi(x)^* dx
0404050504 \lambda = \frac{v}{f}
0439492440 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\right|_0^W
0934990943 k = \frac{2 \pi}{v T}
0948572140 \int \cos(a x) dx = \frac{1}{a}\sin(a x)
1010393913 \langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*
1010393944 x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle
1010923823 k W = n \pi
1020010291 0 = a \sin(k W)
1020394900 p = h/\lambda
1020394902 E = h f
1029039903 p = m v
1029039904 p^2 = m^2 v^2
1158485859 \frac{-\hbar^2}{2m} \nabla^2 = {\cal H}
1202310110 \frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1202312210 \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx
1203938249 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle
1248277773 \cos(2x) = 1-2(\sin(x))^2
1293913110 0 = b
1293923844 \lambda = v T
1314464131 \vec{\nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
1314864131 \vec{\nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}
1395858355 x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle
1492811142 f = x - d  
1492842000 \nabla \vec{x} = f(y)  
1636453295 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = - \nabla^2 \vec{E}
1638282134 \vec{p}_{before} = \vec{p}_{after}
1648958381 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{\nabla}\psi(\vec{r},t)\right)
1857710291 0 = a \sin(n \pi)
1858578388 \nabla^2 E(\vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E(\vec{r})\exp(i \omega t)
1858772113 k = \frac{n \pi}{W}
1934748140 \int |\psi(x)|^2 dx = 1
2029293929 \nabla^2 E(\vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r})\exp(i \omega t)
2103023049 \sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right)
2113211456 f = 1/T
2123139121 -\exp(-i x) = -\cos(x)+i \sin(x)
2131616531 T f = 1
2258485859 {\cal H} \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
2394240499 x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle
2394853829 \exp(-i x) = \cos(-x)+i \sin(-x)
2394935831 ( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
2394935835 \left(\langle\psi| \hat{A} |\psi\rangle \right)^+ = \left(\langle a \rangle\right)^+
2395958385 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{-p^2}{\hbar} \psi(\vec{r},t)
2404934990 \langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
2494533900 \langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
2569154141 \vec{\nabla} \times \frac{\partial}{\partial t}\vec{H} = \epsilon_0 \frac{\partial^2 }{\partial t^2}\vec{E}  
2648958382 \nabla^2 \psi\left(\vec{r},t)\right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p}\psi(\vec{r},t)\right)
2848934890 \langle a \rangle^* = \langle a \rangle
2944838499 \psi(x) = a \sin(\frac{n \pi}{W} x)
3121234211 \frac{k}{2\pi} = \lambda
3121234212 p = \frac{h k}{2\pi}
3121513111 k = \frac{2 \pi}{\lambda}
3131111133 T = 1 / f
3131211131 \omega = 2 \pi f
3132131132 \omega = \frac{2\pi}{T}
3147472131 \frac{\omega}{2 \pi} = f
3285732911 (\cos(x))^2 = 1-(\sin(x))^2
3485475729 \nabla^2 E(\vec{r}) = - \frac{\omega^2}{c^2} E(\vec{r})
3585845894 \langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
3829492824 \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right) = \cos(x)
3924948349 a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0
3942849294 \exp(i x)-\exp(-i x) = 2 i \sin(x)
3943939590 x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle
3947269979 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
3948571256 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi(\vec{r},t)
3948572230 \vec{\nabla}\psi(\vec{r},t) = \psi_0 \vec{\nabla}\exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)  
3948574224 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\vec{k}\cdot\vec{r} - \omega t \right) \right)
3948574226 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)
3948574228 \psi(\vec{r},t) = \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3948574230 \psi(\vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
3948574233 \frac{\partial}{\partial t}\psi(\vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)
3948574235 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}E \psi_0 \exp\left(i\left(\frac{\vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)  
3951205425 \vec{p}_{after} = \vec{p}_{1}
4147472132 E = \frac{h \omega}{2 \pi}
4298359835 E = \frac{1}{2}m v^2
4298359845 E = \frac{1}{2m}m^2 v^2
4298359851 E = \frac{p^2}{2m}
4341171256 i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{p^2}{2 m} \psi(\vec{r},t)
4348571256 \frac{\partial}{\partial t}\psi(\vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi(\vec{r},t)
4394958389 \vec{\nabla}\cdot \left(\vec{\nabla}\psi(\vec{r},t)\right) = \frac{i}{\hbar} \vec{\nabla}\cdot\left( \vec{p} \psi(\vec{r},t)\right)
4585828572 \epsilon_0 \mu_0 = \frac{1}{c^2}
4585932229 \cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
4598294821 \exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
4638429483 \exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))
4742644828 \exp(i x)+\exp(-i x) = 2 \cos(x)
4827492911 \cos(2x)+(\sin(x))^2 = 1-(\sin(x))^2
4838429483 \exp(2 i x) = \cos(2 x)+i \sin(2 x)
4843995999 \frac{1}{2 i}\left(\exp(i x)-\exp(-i x)\right) = \sin(x)
4857472413 1 = \int \psi(x)\psi(x)^* dx
4857475848 \frac{1}{a^2} = \frac{W}{2}
4858429483 \exp(i x)\exp(i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))  
4923339482 i x = \log(y)
4928239482 \log(y) = i x
4928923942 a = b  
4929218492 a+b = c  
4929482992 b = 2
 
4938429482 \exp(-i x) = \cos(x)+i \sin(-x)
4938429483 \exp(i x) = \cos(x)+i \sin(x)
4938429484 \exp(-i x) = \cos(x)-i \sin(x)
4943571230 \vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left(\vec{p}\cdot\vec{r} - E t \right) \right)
4948934890 \langle \psi| \hat{A} |\psi\rangle = \langle a \rangle^*
4949359835 \langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2
4954839242 \cos(2x)+i\sin(2x) = (\cos(x)+i \sin(x))(\cos(x)+i \sin(x))  
4978429483 \exp(-i x) = \cos(-x)+i \sin(-x)  
4984892984 a+2 = c
 
4985825552 \nabla^2 E(\vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E(\vec{r})\exp(i \omega t)
5530148480 \vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}
5727578862 \frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)
5832984291 (\sin(x))^2 + (\cos(x))^2 = 1
5857434758 \int a dx = a x
5868688585 \frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = \frac{p^2}{2m} \psi(\vec{r},t)
5900595848 k = \frac{\omega}{v}
5928285821 x^2 + 2x(b/(2a)) + (b/(2a))^2 = (x+(b/(2a)))^2
5928292841 x^2 + (b/a)x + (b/(2a))^2 = -c/a + (b/(2a))^2
5938459282 x^2 + (b/a)x = -c/a
5958392859 x^2 + (b/a)x+(c/a) = 0
5959282914 x^2 + x(b/a) + (b/(2a))^2 = (x+(b/(2a)))^2
5982958248 x = -\sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
5982958249 x+(b/(2a)) = -\sqrt{(b/(2a))^2 - (c/a)}
5985371230 \vec{\nabla}\psi(\vec{r},t) = \frac{i}{\hbar} \vec{p} \psi(\vec{r},t)
6742123016 \vec{p}_{electron}\cdot\vec{p}_{electron} = (\vec{p}_{1}\cdot\vec{p}_{1})+(\vec{p}_{2}\cdot\vec{p}_{2})-2(\vec{p}_{1}\cdot\vec{p}_{2})
7466829492 \vec{\nabla} \cdot \vec{E} = 0
7564894985 \int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)
7572664728 \cos(2x)+2(\sin(x))^2 = 1
7575738420 \left(\sin\left(\frac{n \pi}{W}x\right)\right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}
7575859295 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859300 \epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{\nabla} \times \vec{E} )_k = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859302 \epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859304 \epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}
7575859306 \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859308 \left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859310 \hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7575859312 \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E} = \vec{\nabla}(\vec{\nabla} \cdot \vec{E} - \nabla^2 \vec{E})
7917051060 \vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}
8139187332 \vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}
8257621077 \vec{p}_{before} = \vec{p}_{1}
8311458118 \vec{p}_{after} = \vec{p}_{2}+\vec{p}_{electron}
8399484849 \langle x^2 -2x\langle x \rangle+\langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2
8484544728 -a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)
8485757728 a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)
8485867742 \frac{2}{W} = a^2
8489593958 d(u v) = u dv + v du
8489593960 d(u v) - v du = u dv
8489593962 u dv = d(u v) - v du
8489593964 \int u dv = u v - \int v du
8494839423 \nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}
8572657110 1 = \int |\psi(x)|^2 dx
8572852424 \vec{E} = E(\vec{r},t)
8575746378 \int \frac{1}{2} dx = \frac{1}{2} x
8575748999 \frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x)\right) = -k^2 \left(a \sin(kx) + b \cos(kx)\right)
8576785890 1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
8577275751 0 = a \sin(0) + b\cos(0)
8582885111 \psi(x) = a \sin(kx) + b \cos(kx)
8582954722 x^2 + 2xh + h^2 = (x+h)^2
8849289982 \psi(x)^* = a \sin(\frac{n \pi}{W} x)
8889444440 1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right)\right)^2 dx
9059289981 \psi(x) = a \sin(k x)
9285928292 ax^2 + bx + c = 0
9291999979 \vec{\nabla} \times \vec{\nabla} \times \vec{E} = -\mu_0\vec{\nabla} \times \frac{\partial \vec{H}}{\partial t}
9294858532 \hat{A}^+ = \hat{A}
9385938295 (x+(b/(2a)))^2 = -(c/a) + (b/(2a))^2
9393939991 \psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9393939992 \psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)
9394939493 \nabla^2 E(\vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E(\vec{r},t)
9429829482 \frac{d}{dx} y = -\sin(x) + i\cos(x)
9482113948 \frac{dy}{y} = i dx
9482438243 (\cos(x))^2 = \cos(2x)+(\sin(x))^2
9482923849 \exp(i x) = y
9482928242 \cos(2x) = (\cos(x))^2 - (\sin(x))^2
9482928243 \cos(2x)+(\sin(x))^2 = (\cos(x))^2
9482943948 \log(y) = i dx
9482948292 b = c
 
9482984922 \frac{d}{dx} y = (i\sin(x) + \cos(x)) i
9483928192 \cos(2x)+i\sin(2x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2
9485384858 \nabla^2 E(\vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E(\vec{r})\exp(i \omega t)
9485747245 \sqrt{\frac{2}{W}} = a
9485747246 -\sqrt{\frac{2}{W}} = a
9492920340 y = \cos(x)+i \sin(x)
9494829190 k  
9495857278 \psi(x=W) = 0
9499428242 E(\vec{r},t) = E(\vec{r})\exp(i \omega t)
9499959299 a + k = b + k  
9582958293 x = \sqrt{(b/(2a))^2 - (c/a)}-(b/(2a))
9582958294 x+(b/(2a)) = \sqrt{(b/(2a))^2 - (c/a)}
9584821911 c + d = a  
9585727710 \psi(x=0) = 0
9596004948 x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle
9848292229 dy = y i dx
9848294829 \frac{d}{dx} y = y i
9858028950 \frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx
9889984281 2(\sin(x))^2 = 1-\cos(2x)
9919999981 \rho = 0
9958485859 \frac{-\hbar^2}{2m} \nabla^2 \psi\left(\vec{r},t)\right) = i \hbar \frac{\partial}{\partial t}\psi(\vec{r},t)
9988949211 (\sin(x))^2 = \frac{1-\cos(2x)}{2}
9991999979 \vec{\nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}
9999998870 \frac{\vec{p}}{\hbar} = \vec{k}
9999999870 \frac{p}{\hbar} = k
9999999950 \beta = 1/(k_{Boltzmann}T)  
9999999951 \langle x | k \rangle = \frac{\exp(i k x)}{\sqrt{2\pi}}  
9999999952 f(x) \delta(x-a) = f(a)
 
9999999953 \int_{-\infty}^{\infty} \delta(x) dx = 1
 
9999999954 c = 1/\sqrt{\epsilon_0 \mu_0}  
9999999955 \vec{E} = -\vec{\nabla} \Psi
 
9999999956 \vec{F} = \frac{\partial}{\partial t}\vec{p}  
9999999957 \vec{F} = -\vec{\nabla} V
 
9999999960 \hbar = h/(2 \pi)
9999999961 \frac{E}{\hbar} = \omega
9999999962 p = \hbar k
9999999963 \lambda = h/p
 
9999999964 \omega = c k  
9999999965 E = \omega \hbar
9999999966 \vec{L} = \vec{r}\times\vec{p}
 
9999999967 \vec{S} = \frac{1}{\mu_0} \vec{E}\times \vec{B}
 
9999999968 x = \frac{-b-\sqrt{b^2-4ac}}{2a}
9999999969 x = \frac{-b+\sqrt{b^2-4ac}}{2a}
9999999970 \eta_1 \sin\theta_1 = \eta_2 \sin\theta_2
 
9999999971 {\cal H} = \frac{p^2}{2m}+V
 
9999999972 {\cal H} |\psi\rangle = E |\psi\rangle
 
9999999973 \left( \Delta A \right)^2 = \langle A^2 \rangle - \langle A \rangle^2
 
9999999974 \langle \psi| \hat{A} |\psi\rangle = \int_{-\infty}^{\infty} \psi^* A \psi dx
 
9999999975 \langle \psi| \hat{A} |\psi\rangle = \langle a \rangle
9999999976 \hat{p} = -i \hbar \frac{\partial }{\partial x}  
9999999977 [\hat{x},\hat{p}] = i \hbar  
9999999978 \vec{\nabla} \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}
 
9999999979 \vec{\nabla} \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}  
9999999980 \vec{\nabla} \cdot \vec{B} = 0
 
9999999981 \vec{\nabla} \cdot \vec{E} = \rho/\epsilon_0
9999999982 V = I R + Q/C + L \frac{\partial I}{\partial t}  
9999999983 C = V A/d
 
9999999984 Q = C V
 
9999999985 V = I R
 
9999999986 \left[cos⁡θsin⁡θ−sin⁡θcos⁡θ\right]\left[AxAy\right] = \left[Ax′Ay′\right]
 
9999999987 1 atmosphere = 101.325 Pascal
 
9999999988 1 atmosphere = 14.7 pounds/(inch^2)
 
9999999989 mass_{electron} = 511000 electronVolts/(q^2)
 
9999999990 Tesla = 10000*Gauss
 
9999999991 Pascal = Newton / (meter^2)
 
9999999992 Tesla = Newton / (Ampere*meter)
 
9999999993 Farad = Coulumb / Volt
 
9999999995 Volt = Joule / Coulumb
 
9999999996 Coulomb = Ampere / second
 
9999999997 Watt = Joule / second
 
9999999998 Joule = Newton*meter
 
9999999999 Newton = kilogram*meter/(second^2)
 
4961662865 x
9110536742 2 x
8483686863 \sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x)\right)
3470587782 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \frac{1}{2}\left(\exp(i x)+\exp(-i x)\right)
8642992037 2
9894826550 2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x)\right) \left(\exp(i x)+\exp(-i x)\right)
4257214632 \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)  
8699789241 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)
9180861128 2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)
2405307372 \sin(2 x) = 2 \sin(x) \cos(x)
3268645065 x
9350663581 \pi
8332931442 \exp(i \pi) = \cos(\pi)+i \sin(\pi)
6885625907 \exp(i \pi) = -1 + i 0
3331824625 \exp(i \pi) = -1
4901237716 1
2501591100 \exp(i \pi) + 1 = 0
5136652623 E = KE + PE mechanical energy is the sum of the potential plus kinetic energies
7915321076 E, KE, PE  
3192374582 E_2, KE_2, PE_2
 
7875206161 E_2 = KE_2 + PE_2
4229092186 E, KE, PE  
1490821948 E_1, KE_1, PE_1
 
4303372136 E_1 = KE_1 + PE_1
5514556106 E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)
8357234146 KE = \frac{1}{2} m v^2 kinetic energy Kinetic_energy
3658066792 KE_2, v_2
 
8082557839 KE, v  
7676652285 KE_2 = \frac{1}{2} m v_2^2
2790304597 KE_1, v_1
 
9880327189 KE, v  
4928007622 KE_1 = \frac{1}{2} m v_1^2
5733146966 KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)
6554292307 t
4270680309 \frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}
5781981178 x^2 - y^2 = (x+y)(x-y) difference of squares Difference_of_two_squares
5946759357 v_2, v_1
 
6675701064 x, y  
4648451961 v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)
9356924046 \frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}
9397152918 v = \frac{v_1 + v_2}{2} average velocity
2857430695 a = \frac{v_2 - v_1}{t}
acceleration
7735737409 \frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}
4784793837 \frac{KE_2 - KE_1}{t} = m v a
5345738321 F = m a Newton's second law of motion Newton%27s_laws_of_motion#Newton's_second_law
2186083170 \frac{KE_2 - KE_1}{t} = v F
6715248283 PE = -F x potential energy Potential_energy
3852078149 PE_2, x_2
 
7388921188 PE, x  
2431507955 PE_2 = -F x_2
3412641898 PE_1, x_1
 
9937169749 PE, x  
4669290568 PE_1 = -F x_1
7734996511 PE_2 - PE_1 = -F ( x_2 - x_1 )
2016063530 t
7267155233 \frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)
9337785146 v = \frac{x_2 - x_1}{t} average velocity
4872970974 \frac{PE_2 - PE_1}{t} = -F v
2081689540 t
2770069250 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}
3591237106 \frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v
1772416655 \frac{E_2 - E_1}{t} = v F - F v
1809909100 \frac{E_2 - E_1}{t} = 0
5778176146 t
3806977900 E_2 - E_1 = 0
5960438249 E_1
8558338742 E_2 = E_1
8945218208 \theta_{Brewster} + \theta_{refracted} = 90^{\circ} based on figure 34-27 on page 824 in 2001_HRW
9025853427 \theta_{Brewster}
1310571337 \theta_{refracted} = 90^{\circ} - \theta_{Brewster}
6450985774 n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 ) Law of Refraction eq 34-44 on page 819 in 2001_HRW
1779497048 \theta_{Brewster}, \theta_{refraction}  
7322344455 \theta_1, \theta_2
 
2575937347 n_1 \sin( \theta_{Brewster} ) = n_2 \sin( \theta_{refraction} )
7696214507 n_1 \sin( \theta_{Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{Brewster} )
8588429722 \sin( 90^{\circ} - x ) = \cos( x )
7375348852 \theta_{Brewster}
1512581563 x
6831637424 \sin( 90^{\circ} - \theta_{Brewster} ) = \cos( \theta_{Brewster} )
3061811650 n_1 \sin( \theta_{Brewster} ) = n_2 \cos( \theta_{Brewster} )
7857757625 n_1
9756089533 \sin( \theta_{Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{Brewster} )
5632428182 \cos( \theta_{Brewster} )
2768857871 \frac{\sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )} = \frac{n_2}{n_1}
4968680693 \tan( x ) = \frac{ \sin( x )}{\cos( x )}
7321695558 \theta_{Brewster}
9906920183 x
4501377629 \tan( \theta_{Brewster} ) = \frac{ \sin( \theta_{Brewster} )}{\cos( \theta_{Brewster} )}
3417126140 \tan( \theta_{Brewster} ) = \frac{ n_2 }{ n_1 }
5453995431 \arctan{ x }
6023986360 x
8495187962 \theta_{Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }
9040079362 f
9565166889 T
5426308937 v = \frac{d}{t}
7476820482 C
1277713901 d
6946088325 v = \frac{C}{t}
6785303857 C = 2 \pi r
4816195267 C, r
 
2113447367 C_{\rm{Earth\ orbit}}, r_{\rm{Earth\ orbit}}
 
6348260313 C_{\rm{Earth\ orbit}} = 2 \pi r_{\rm{Earth\ orbit}}
3847777611 C, v, t  
6406487188 C_{\rm{Earth\ orbit}}, v_{\rm{Earth\ orbit}}, t_{\rm{Earth\ orbit}}
 
3046191961 v_{\rm{Earth\ orbit}} = \frac{C_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
3080027960 v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{t_{\rm{Earth\ orbit}}}
7175416299 t_{\rm{Earth\ orbit}} = 1 {\rm{year}}
3219318145 \frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}
8721295221 t_{\rm{Earth\ orbit}} = 3.16 10^7 {\rm{seconds}}
4593428198 v_{\rm{Earth\ orbit}} = \frac{2 \pi r_{\rm{Earth\ orbit}}}{3.16\ 10^7 {\rm seconds}}
3472836147 r_{\rm{Earth\ orbit}} = 1.496\ 10^8 {\rm km}
6998364753 v_{\rm{Earth\ orbit}} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}
4180845508 v_{\rm{Earth\ orbit}} = 29.8 \frac{{\rm km}}{{\rm sec}}
4662369843 x' = \gamma (x - v t)
2983053062 x = \gamma (x' + v t')
3426941928 x = \gamma ( \gamma (x - v t) + v t' )
2096918413 x = \gamma ( \gamma x - \gamma v t + v t' )
7741202861 x = \gamma^2 x - \gamma^2 v t + \gamma v t'
7337056406 \gamma^2 x
4139999399 x - \gamma^2 x = - \gamma^2 v t + \gamma v t'
3495403335 x
9031609275 x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'
8014566709 \gamma^2 v t
9409776983 x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'
2226340358 \gamma v
6417705759 \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = \gamma v t'
 
8730201316 \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'
first term was multiplied by \gamma/\gamma
1974334644 \frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'
5148266645 t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t
4287102261 x^2 + y^2 + z^2 = c^2 t^2 describes a spherical wavefront
1201689765 x'^2 + y'^2 + z'^2 = c^2 t'^2
describes a spherical wavefront for an observer in a moving frame of reference
7057864873 y' = y
frame of reference is moving only along x direction
8515803375 z' = z
frame of reference is moving only along x direction
6153463771 3
 
4746576736 asdf
 
1027270623 minga
 
6101803533 ming
 
9231495607 a = b
 
8664264194 b + 2
 
9805063945 \gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2
4057686137 C \to C_{\rm{Earth\ orbit}}
2346150725 r \to r_{\rm{Earth\ orbit}}
9753878784 v \to v_{\rm{Earth\ orbit}}
8135396036 t \to t_{\rm{Earth\ orbit}}
2773628333 \theta_1 \to \theta_{Brewster}
7154592211 \theta_2 \to \theta_{\rm refracted}
3749492596 E \to E_1
1258245373 E \to E_2
4147101187 KE \to KE_1
3809726424 PE \to PE_1
6383056612 KE \to KE_2
5075406409 PE \to PE_2
5398681503 v \to v_1
9305761407 v \to v_2
4218009993 x \to x_1
4188639044 x \to x_2
8173074178 x \to v_2
1025759423 y \to v_1
1935543849 \gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2
1586866563 \left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)
3182633789 \gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1
1916173354 -\gamma^2 v^2 + c^2 \gamma^2 = c^2
2076171250 -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0
5284610349 \gamma^2
2417941373 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2
5787469164 1 - \gamma^2
1639827492 - c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1
5669500954 v^2 \gamma^2
5763749235 -c^2 + c^2 \gamma^2 = v^2 \gamma^2
6408214498 c^2
2999795755 c^2 \gamma^2 = v^2 \gamma^2 + c^2
3412946408 v^2 \gamma^2
2542420160 c^2 \gamma^2 - v^2 \gamma^2 = c^2
7743841045 \gamma^2
7513513483 \gamma^2 (c^2 - v^2) = c^2
8571466509 c^2 - \gamma^2
7906112355 \gamma^2 = \frac{c^2}{c^2 - \gamma^2}
3060139639 1/2
 
1528310784 \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
8360117126 \gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}} not a physically valid result in this context
3366703541 a = \frac{v - v_0}{t} acceleration is the average change in speed over a duration
7083390553 t
4748157455 a t = v - v_0
6417359412 v_0
4798787814 a t + v_0 = v
3462972452 v = v_0 + a t
3411994811 v_{\rm ave} = \frac{d}{t}
6175547907 v_{\rm ave} = \frac{v + v_0}{2}
9897284307 \frac{d}{t} = \frac{v + v_0}{2}
8865085668 t
8706092970 d = \left(\frac{v + v_0}{2}\right)t
7011114072 d = \frac{(v_0 + a t) + v_0}{2} t
1265150401 d = \frac{2 v_0 + a t}{2} t
9658195023 d = v_0 t + \frac{1}{2} a t^2
5799753649 2
7215099603 v^2 = v_0^2 + 2 a t v_0 + a^2 t^2
5144263777 v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)
7939765107 v^2 = v_0^2 + 2 a d
6729698807 v_0
9759901995 v - v_0 = a t
2242144313 a
1967582749 t = \frac{v - v_0}{a}
5733721198 d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)
5611024898 d = \frac{1}{2 a} (v^2 - v_0^2)
5542390646 2 a
8269198922 2 a d = v^2 - v_0^2
9070454719 v_0^2
4948763856 2 a d + v_0^2 = v^2
9645178657 a t
6457044853 v - a t = v_0
1259826355 d = (v - a t) t + \frac{1}{2} a t^2
4580545876 d = v t - a t^2 + \frac{1}{2} a t^2
6421241247 d = v t - \frac{1}{2} a t^2
4182362050 Z = |Z| \exp( i \theta ) Z \in \mathbb{C}
1928085940 Z^* = |Z| \exp( -i \theta )
9191880568 Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )
3350830826 Z Z^* = |Z|^2
8396997949 I = | A + B |^2 intensity of two waves traveling opposite directions on same path
4437214608 Z
5623794884 A + B
2236639474 (A + B)(A + B)^* = |A + B|^2
1020854560 I = (A + B)(A + B)^*
6306552185 I = (A + B)(A^* + B^*)
8065128065 I = A A^* + B B^* + A B^* + B A^*
9761485403 Z
8710504862 A
4075539836 A A^* = |A|^2
6529120965 B
1511199318 Z
7107090465 B B^* = |B|^2
5125940051 I = |A|^2 + B B^* + A B^* + B A^*
1525861537 I = |A|^2 + |B|^2 + A B^* + B A^*
1742775076 Z \to B
7607271250 \theta \to \phi
4192519596 B = |B| \exp(i \phi)
2064205392 A
1894894315 Z
1357848476 A = |A| \exp(i \theta)
6555185548 A^* = |A| \exp(-i \theta)
4504256452 B^* = |B| \exp(-i \phi)
7621705408 I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)
3085575328 I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))
4935235303 x
2293352649 \theta - \phi
3660957533 \cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
3967985562 2
2700934933 2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)
8497631728 I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )
6774684564 \theta = \phi for coherent waves
2099546947 I = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )  
2719691582 |A| = |B| in a loop
3257276368 I = |A|^2 + |A|^2 + |A| |A| 2
 
8283354808 I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )
8046208134 I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2
1172039918 I_{\rm coherent} = 4 |A|^2
8602221482 \langle \cos(\theta - \phi) \rangle = 0 incoherent light source
6240206408 I_{\rm incoherent} = |A|^2 + |B|^2
6529793063 I_{\rm incoherent} = |A|^2 + |A|^2
3060393541 I_{\rm incoherent} = 2|A|^2
6556875579 \frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2
8750500372 f
 
7543102525 g
 
8267133829 a = b
 
7968822469 c = d
 
3169580383 \vec{a} = \frac{d\vec{v}}{dt} acceleration is the change in speed over a duration
8602512487 \vec{a} = a_x \hat{x} + a_y \hat{y} decompose acceleration into two components
4158986868 a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}
5349866551 \vec{v} = v_x \hat{x} + v_y \hat{y}
7729413831 a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)
1819663717 a_x = \frac{d}{dt} v_x
8228733125 a_y = \frac{d}{dt} v_y
9707028061 a_x = 0
2741489181 a_y = -g
3270039798 2
8880467139 2
8750379055 0 = \frac{d}{dt} v_x
1977955751 -g = \frac{d}{dt} v_y
6672141531 dt
1702349646 -g dt = d v_y
8584698994 -g \int dt = \int d v_y
9973952056 -g t = v_y - v_{0, y}
4167526462 v_{0, y}
6572039835 -g t + v_{0, y} = v_y
7252338326 v_y = \frac{dy}{dt}
6204539227 -g t + v_{0, y} = \frac{dy}{dt}
1614343171 dt
8145337879 -g t dt + v_{0, y} dt = dy
8808860551 -g \int t dt + v_{0, y} \int dt = \int dy
2858549874 - \frac{1}{2} g t^2 + v_{0, y} t = y - y_0
6098638221 y_0
2461349007 - \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y
8717193282 dt
1166310428 0 dt = d v_x
2366691988 \int 0 dt = \int d v_x
1676472948 0 = v_x - v_{0, x}
1439089569 v_{0, x}
6134836751 v_{0, x} = v_x
8460820419 v_x = \frac{dx}{dt}
7455581657 v_{0, x} = \frac{dx}{dt}
8607458157 dt
1963253044 v_{0, x} dt = dx
3676159007 v_{0, x} \int dt = \int dx
9882526611 v_{0, x} t = x - x_0
3182907803 x_0
8486706976 v_{0, x} t + x_0 = x
1306360899 x = v_{0, x} t + x_0
7049769409 2
9341391925 \vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}
6410818363 \theta
7391837535 \cos(\theta) = \frac{v_{0, x}}{v_0}
7376526845 \sin(\theta) = \frac{v_{0, y}}{v_0}
5868731041 v_0
6083821265 v_0 \cos(\theta) = v_{0, x}
5438722682 x = v_0 t \cos(\theta) + x_0
5620558729 v_0
8949329361 v_0 \sin(\theta) = v_{0, y}
1405465835 y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0
9862900242 y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0
6050070428 v_{0, x}
3274926090 t = \frac{x - x_0}{v_{0, x}}
7354529102 y = - \frac{1}{2} g \left( \frac{x - x_0}{v_{0, x}} \right)^2 + v_{0, y} \frac{x - x_0}{v_{0, x}} + y_0
8406170337 y \to y_f
2403773761 t \to t_f
5379546684 y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
5373931751 t = t_f
9112191201 y_f = 0
8198310977 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0
1650441634 y_0 = 0
define coordinate system such that initial height is at origin
1087417579 0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)
4829590294 t_f
2086924031 0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)
6974054946 \frac{1}{2} g t_f
1191796961 \frac{1}{2} g t_f = v_0 \sin(\theta)
2510804451 2/g
4778077984 t_f = \frac{2 v_0 \sin(\theta)}{g}
3273630811 x \to x_f
6732786762 t \to t_f
3485125659 x_f = v_0 t_f \cos(\theta) + x_0
4370074654 t = t_f
2378095808 x_f = x_0 + d
4268085801 x_0 + d = v_0 t_f \cos(\theta) + x_0
8072682558 x_0
7233558441 d = v_0 t_f \cos(\theta)
2297105551 d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)
7587034465 \theta
7214442790 x
2519058903 \sin(2 \theta) = 2 \sin(\theta) \cos(\theta)
8922441655 d = \frac{v_0^2}{g} \sin(2 \theta)
5667870149 \theta
1541916015 \theta = \frac{\pi}{4}
3607070319 d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)
5353282496 d = \frac{v_0^2}{g}
Physics Derivation Graph: 6580 Expressions

Delete an expression

 This action requires you to be signed in

Edit the name of an expression

 This action requires you to be signed in

Add a symbol to the AST for an expression

 This action requires you to be signed in

Remove symbol from the AST for an expression

 This action requires you to be signed in

Edit the latex for an expression

 This action requires you to be signed in

Edit the note for an expression

 This action requires you to be signed in